STAT7310 and STAT8310: Statistical Theory
Estimation and Estimators
Semester 1, 2021
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/ Estimation and estimators
Estimation and estimators
I Suppose we have a random sample X where element Xi i.i.d.∼ fX (·; θ)
for some density or pf fX (·; θ)
I f is assumed known but depends on the parameter θ ∈ Θ.
I Θ is the set of possible values of θ (called the parameter space).
I The parameter could be a scalar.
I e.g. Xi ∼ Poisson(λ) distribution
I θ = λ ∈ R+ = Θ
I The parameter could be a vector. E.g.
I Xi ∼ Beta(α, β) which requires α > 0 and β > 0.
I θ =
(
θ1 θ2
)T ∈ {(x y)T : x ∈ R+, y ∈ R+,} = Θ
I Xi ∼ N (µ, σ2) which requires µ ∈ R and σ > 0.
I θ =
(
θ1 θ2
)T ∈ {(x y)T : x ∈ R, y ∈ R+,} = Θ
I Two scenarios
I θ is known, then the pdf or pf is completely specified. (nothing to do)
I θ is not (completely) known, we are faced with the practical problem
of estimating it.
DEPARTMENT OF MATHEMATICS & STATISTICS 3
Joint pdf for i.i.d. random variables
I If the random variables X =
(
X1 X2 . . . Xn
)T are i.i.d. with
common pdf or pf fX (x ; θ) , their joint pdf or pf is
fX1,X2,...,Xn (x1, x2, . . . , xn; θ) =
n∏
i=1
fX (xi ; θ) .
I The most basic principle of statistical inference is that we must
estimate θ using only our data
{x1, x2, . . . , xn} ,
which is assumed to constitute a single observation on
{X1,X2, . . . ,Xn} .
I There are essentially two approaches:
1. Point Estimation;
2. Interval Estimation
I Recall the definition of a statistic:
t (X1,X2, . . . ,Xn) = t (X)
is a function of the random sample X =
(
X1 X2 · · · Xn
)T
DEPARTMENT OF MATHEMATICS & STATISTICS 4
Point Estimation
I Recall
I The random sample, X =
(
X1 X2 . . . Xn
)T and,
I The observed data, x =
(
x1 x2 . . . xn
)T .
I We may use a statistic t (X) to estimate θ
I t (X) is then called a point estimator of θ.
I Usually denoted by θ̂ = t(X).
I tx is called a point estimate of θ.
I Principle can be extended to estimate τ(θ) some function of θ. That
is, a different t (X) can be considered to estimate a function of θ
I Even when θ is a single scalar, it is sometimes desirable to be able to
estimate τ(θ), a function of θ.
DEPARTMENT OF MATHEMATICS & STATISTICS 5
Example: Two Point estimators
I Suppose X ∼ Exp (θ)
f (x ; θ) =
{
1
θ e−
x
θ , x ≥ 0;
0, x < 0.
I Then E (X) = θ and var(X) = θ2.
I To estimate θ, we might then use
t1(X) = t1 (X1,X2, . . . ,Xn) = X =
1
n
n∑
i=1
Xi
or
t2(X) = t2 (X1,X2, . . . ,Xn) =
√
S2n =
√√√√1
n
n∑
i=1
(
Xi − X
)2
to estimate θ
I How do we choose an estimator in general? We must have some
methods for computing estimators, as well as methods for comparing
them.
DEPARTMENT OF MATHEMATICS & STATISTICS 6
Interval Estimation
I Again, define two statistics
t1(X) = t1 (X1,X2, . . . ,Xn)
t2(X) = t2 (X1,X2, . . . ,Xn)
where t1 (X) < t2 (X) are such that
(t1 (X) , t2 (X))
is a random interval, which we hope includes the true value of θ
I We may consider (t1 (X) , t2 (X)) as an interval estimator of θ
I Usually constructed in such a way that there is a certain probability
that the interval estimator contains θ
I Recall confidence intervals!
I This is considered in detail in the confidence intervals topic.
DEPARTMENT OF MATHEMATICS & STATISTICS 7
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/ Methods for Generating
Estimators
The Method of Moments
I Suppose the pdf or pf of each of the i.i.d. Xi is
fX (x ;θ)
where θ =
(
θ1 θ2 · · · θk
)T
.
I The r th raw moment is µ′r = E (X r ) which is a function of the k
parameters, i.e. µ′r = µ′r (θ1, θ2, . . . , θk)
I The sample r th (raw) moment is M ′r = 1n
n∑
i=1
X ri
I The sample moment can be used to estimate the population moment:
solve for θ1, θ2, . . . , θk the equations
M ′r = µ′r (θ1, θ2, . . . , θk) ; r = 1, 2, . . . , k
I These are k equations in k unknowns which (subject to some
conditions) will result in a unique solution.
I The solutions form the “method of moments” estimators of
θ1, θ2, . . . , θk .
DEPARTMENT OF MATHEMATICS & STATISTICS 9
Method of Moments Example 1
I Suppose Xi i.i.d∼ B (n, p), and we wish to estimate θ = p using the
method of moments.
I The first moment is
µ′1 = E (X) = np.
I The sample moments are
M′1 =
1
n
n∑
i=1
Xi = X
I Solving yields:
µ′1(p̂) = M′1
⇓
np̂ = 1n
n∑
i=1
Xi = X
p̂mom =
X
n
I Here X = average number of successes ⇒ X/n = average proportion
of successes.
DEPARTMENT OF MATHEMATICS & STATISTICS 10
Method of Moments Example 2
I Suppose Xi , i.i.d.∼ N
(
µ, σ2
)
, and we wish to estimate θ =
(
µ σ2
)T
using the method of moments.
I The population moments are
µ′1 = E (X) = µ and µ′2 = E
(
X 2
)
= µ2 + σ2.
I The sample moments are
M′1 = X and M′2 =
1
n
n∑
i=1
X 2i .
I Solve simultaneously:
µ′1(µ̂mom, σ̂2mom) = M′1
µ′2(µ̂mom, σ̂2mom) = M2;
⇓
µ̂mom = X
(µ̂mom)2 + σ̂2mom =
1
n
n∑
i=1
X 2i
DEPARTMENT OF MATHEMATICS & STATISTICS 11
I The first equation is already solves µ̂mom = X .
I The second equation requires some algebra are
σ̂2mom =
1
n
n∑
i=1
X 2i − µ̂mom
= 1n
n∑
i=1
X 2i − X
2
= 1n
n∑
i=1
X 2i − 2X
2 + X 2
= 1n
n∑
i=1
X 2i − 2X
1
n
n∑
i=1
Xi +
1
n
n∑
i=1
X 2
= 1n
n∑
i=1
(
X 2i − 2XXi + X
2)
= 1n
n∑
i=1
(
Xi − X
)2
.
DEPARTMENT OF MATHEMATICS & STATISTICS 12
Method of moments Example 3
I Suppose X1,X2, . . . ,Xn are i.i.d. U [0, θ] .
fX (x ; θ) =
{
1
θ , 0 ≤ x ≤ θ;
0, otherwise.
I Computing the first moment,
µ′1 = E (X) =
∫ θ
0
x
θ
dx = θ2 .
I Sample moment: M′1 = X .
I Method of moments estimator of θ is 2X .
I NB: This does not always lead to feasible estimators.
I e.g. Suppose
x = (x1, x2, x3) = (0.1, 0.3, 7.4)
then the moment estimate of θ is
2x = 2× (0.1 + 0.3 + 7.4) /3 = 5.2
I However all data must be less than θ, and x3 = 7.4 > 5.2 = θ̂.
I Sometimes method of moment estimators aren’t sensible.
DEPARTMENT OF MATHEMATICS & STATISTICS 13
The Likelihood Function
I Maximum likelihood estimation is probably the most important
concept in all of Statistical Inference.
I Suppose X1,X2, . . . ,Xn have joint pdf or pf fX (x ; θ) . It is not
actually necessary that the Xi be independent or identically
distributed. We just need their joint pdf or pf.
I The likelihood, or likelihood function, is thought of as a function of θ.
The likelihood function is defined by
L (θ;X ) = fX (x ; θ)|x=X .
Note that L is a random function, since it depends on the random
vector X . To construct L we form the joint pdf or pf of X at a value
x of X , and then replace x by X .
I If the Xi are i.i.d. with common pdf or pf f (x ; θ) , then
L (θ;X) =
n∏
i=1
f (xi ; θ)|xi=Xi
DEPARTMENT OF MATHEMATICS & STATISTICS 14
The Likelihood Function: Example 1
I {Xi} i.i.d. Exponential with mean θ. Then
f (x ; θ) =
{
1
θ e−
x
θ , x ≥ 0
0; x < 0,
and the joint pdf is
fX (x ; θ) =

n∏
i=1
(
1
θ e−
xi
θ
)
, x1, . . . , xn ≥ 0;
0, otherwise.
=
{
θ−n exp
(−∑ni=1 xi/θ) , x1, . . . , xn ≥ 0
0, otherwise.
I The likelihood is
L (θ;X) =
{
θ−n exp
(−∑ni=1 Xi/θ) , θ > 0
0, otherwise.
I NB: the domain of L is different to fX , as they are functions of
different quantities.
DEPARTMENT OF MATHEMATICS & STATISTICS 15
The Likelihood Function: Example 2
I {Xi} i.i.d. Poisson with mean θ. Then the common pf is
f (x ; θ) =
{
e−θθx
x ! , x = 0, 1, . . .
0, otherwise.
Thus noting that
∑n
i=1 xi = nx
fX (x ; θ) =

n∏
i=1
(
e−θθxi
xi !
)
, x1, . . . , xn ∈ {0, 1, . . .}
0, otherwise.
=
e−nθ θnx/
n∏
i=1
xi !, x1, . . . , xn ∈ {0, 1, . . .}
0, otherwise.
and the likelihood is
L (θ;X ) =
e−nθ θnX/
n∏
i=1
Xi !, θ > 0
0, otherwise.
DEPARTMENT OF MATHEMATICS & STATISTICS 16
Maximum Likelihood Estimators (MLEs) I
I To compute the MLE
1. Form the joint pdf or pf and thus the likelihood.
2. Maximise the likelihood.
I It is sometimes better to maximise the log likelihood.
I Why? Since the maximiser of a function is the same as the maximiser
of any monotonic increasing function of that function.
I Clearly L and L = log L are maximised at the same values.
I As L is usually a product, L is usually a sum and is easier to maximise.
I L is called the log-likelihood.
I Sometimes calculus is used to maximise L.
I Find stationary points θ̂ such that ∂L(θ)
∂θ
= 0
I Check the stationary point is a maxima by checking concavity. In the
Single parameter case: ∂2L
∂θ2 < 0 at θ = θ̂. In the multi parameter case:
The matrix of derivatives ∂2L
∂θi∂θj
is negative-definite at θ = θ̂.
I Other times common sense must be used. We’ll see examples of both
cases.
DEPARTMENT OF MATHEMATICS & STATISTICS 17
MLE Example 1
1. Suppose {Xi} i.i.d. Bernoulli(p) .
I Then the common pf is
f (x ; p) =

p, x = 1
1− p, x = 0
0, otherwise.
=
{
px (1− p)1−x , x = 0, 1
0, otherwise.
Thus, as long as the xi are all 0 or 1,
fX (x ; p) =
n∏
i=1
{
pxi (1− p)1−xi
}
= p
∑n
i=1
xi (1− p)
∑n
i=1
(1−xi )
= p
∑n
i=1
xi (1− p)n−
∑n
i=1
xi
= pnx (1− p)n(1−x)
DEPARTMENT OF MATHEMATICS & STATISTICS 18
I Hence
L (p;X ) =
{
pnX (1− p)n(1−X) , 0 < p < 1
0, otherwise.
and
L (p;X ) = nX log p + n (1− X) log (1− p)
= n
{
X log p +
(
1− X) log (1− p)} .
I The MLE of p is found by maximising L with respect to p, since L is
difficult to maximise directly. Calculus is clearly needed. Now
∂L
∂p = n
(
X
p −
1− X
1− p
)
= np (1− p)
{
X (1− p)− p (1− X)}
= np (1− p)
{
X − pX − p + pX}
= np (1− p)
{
X − p} .
Hence ∂L∂p = 0 when p = p̂MLE = X . Also
∂2L
∂p2 = n
(
− Xp2 −
1− X
(1− p)2
)
< 0,∀p.
DEPARTMENT OF MATHEMATICS & STATISTICS 19
I Thus L is uniquely maximised when p = X , then,
I the MLE of p is p̂ = X .
I Note that p̂ is a random variable, not a constant.
I We can thus talk about the distribution of the MLE.
I Note that 0 ≤ X ≤ 1
I if X = 0 or 1, we are in trouble if we use this analysis!
I it is left as an exercise to discover the MLE of p when X = 0 or 1.
DEPARTMENT OF MATHEMATICS & STATISTICS 20
MLE : Example 2
2. Xi i.i.d.∼ N
(
µ, σ2
)
.
I Define θ =
(
µ σ2
)T
, the common pdf is
f (x ; θ) = 1√
2piσ2
e−
(x−µ)2
2σ2
and the joint pdf is
fX (x ; θ) =
n∏
i=1
{
1√
2piσ2
e−
(xi−µ)2
2σ2
}
= 1
(2piσ2)
n
2
exp
{
−
n∑
i=1
(xi − µ)2
2σ2
}
.
I Thus, as long as σ2 > 0,
L (θ;X ) = 1
(2piσ2)
n
2
exp
{
−
n∑
i=1
(Xi − µ)2
2σ2
}
and
L (θ;X ) = −n2 log
(
2piσ2
)− 12σ2
n∑
i=1
(Xi − µ)2 .
DEPARTMENT OF MATHEMATICS & STATISTICS 21
I L is function of two variables, µ and σ2. Now
∂L
∂µ
= 1
σ2
n∑
i=1
(Xi − µ)
which is clearly 0 only when µ = X .
I Hence the MLE of µ is µ̂ = X . Also
∂L
∂σ2
= − n2σ2 +
1
2 (σ2)2
n∑
i=1
(Xi − µ)2
= − n
2 (σ2)2
{
σ2 − 1n
n∑
i=1
(Xi − µ)2
}
.
I This is 0 when σ2 = 1n
∑n
i=1 (Xi − µ)2 .
I Is this then the MLE of σ2?
I No. It is not even an estimator, as it depends on µ.
DEPARTMENT OF MATHEMATICS & STATISTICS 22
I The MLE of θ is the value of θ which maximises L.
I If there is a value of θ at which the two partial derivatives of L are
both zero, then this will be the MLE.
I In this case, we also must have µ = X , so the MLE of σ2 is
σ̂2 = 1n
∑n
i=1
(
Xi − X
)2
, which is the moment estimator of σ2.
DEPARTMENT OF MATHEMATICS & STATISTICS 23
MLE : Further examples
3. {Xi} i.i.d. and exponentially distributed with mean θ. From earlier
L (θ;X ) = −n log θ −
∑n
i=1 Xi
θ
.
∂L
∂θ
= −n
θ
+
∑n
i=1 Xi
θ2
= 0
when θ = X . Thus θ̂ = X is the MLE of θ.
4. {Xi} i.i.d. Poisson with mean θ. Then, from above
L (θ;X ) = −nθ +
n∑
i=1
Xi log θ − log
( n∏
i=1
Xi !
)
.
∂L
∂θ
= −n +
∑n
i=1 Xi
θ
= 0.
when θ = X . Thus θ̂ = X is the MLE of θ.
DEPARTMENT OF MATHEMATICS & STATISTICS 24
MLE Domain and boundary example
5. {Xi} i.i.d. U (a, b) . The joint pdf is
fX (x ; θ) =
{
1
(b−a)n , x1, x2, . . . , xn ∈ (a, b)
0, otherwise.
NB: x1, x2, . . . , xn ∈ (a, b) is equivalent to a < x(1) and b > x(n).
⇒ L (θ;X ) =
{
−n log (b − a) , a < X(1) and b > X(n)
−∞, otherwise.
I Problems arising here.
I Cannot find the MLEs of a and b by differentiation.
I L is made as large as possible by making b − a as small as possible.
I Since a < X(1) and b > X(n), the difference b − a > X(n) − X(1).
I However, b cannot equal X(n) and a cannot equal X(1).
I There is no MLE! The MLE doesn’t work here?
I Consider instead, Xi i.i.d.∼ U [a, b] ,
I As the random variables are continuous, this makes no difference.
I Same mathematical formulation holds with strict inequalities replaced
with non-strict.
I Now there are MLEs: the MLE of a is X(1) and the MLE of b is X(n).
DEPARTMENT OF MATHEMATICS & STATISTICS 25
MLE Comments
I Notes:
1. The domain and boundary problems of the last example are resolved
using the concept of supremum instead of maximum.
2. Putting the individual partial derivatives to 0 rarely gives the MLE of
a component of θ unless θ is 1-dimensional.
3. The 2pi term vanishes in the definition of L for the normal
distribution. Make sure you understand why.
4. Note that we differentiate with respect to σ2, and not σ, for the
normal case. See what happens if you think of σ as a parameter,
instead of σ2.
DEPARTMENT OF MATHEMATICS & STATISTICS 26
//
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/ Properties of Estimators
Bias and Unbiasedness
I Let X have joint pdf or pf fX (x; θ) and let T = t (X) be an
estimator of τ (θ) .
Definition
T is an unbiased estimator of τ (θ) if E (T ) = τ (θ) , ∀θ ∈ Θ Note that
expectations are computed assuming that fX (x ; θ) is the joint pdf or pf
(See below).
Definition
The bias in estimating τ (θ) by T is given by bias(T ) = E (T )− τ (θ) .
I Note: the expectation of T is given (assuming that X is continuous)
by
E (T ) =
∫ ∞
−∞
· · ·
∫ ∞
−∞
t (x1, . . . , xn) fX (x1, . . . , xn; θ) dx1dx2 . . . dxn,
which is, of course, a function of θ alone.
DEPARTMENT OF MATHEMATICS & STATISTICS 28
Examples I
I X is an unbiased estimator of E (Xi) as long as all of the E (Xi) are
the same. Indeed,
E
(
X
)
= 1nE
( n∑
i=1
Xi
)
= 1nnE (X1)
= E (X1) .
I Another example: Let X1,X2, . . . ,Xn be i.i.d. with common pdf
f (x ; θ) =
{
θe−θx , x > 0;
0, x ≤ 0.
I Then E (X1) = 1θ and so X is an unbiased estimator of
1
θ
.
I In this case, is 1
X
an unbiased estimator of θ?
I It can be shown that Y =
∑n
i=1 Xi has pdf
fY (y ; θ) =
{
θnyn−1e−θy
(n−1)! , y > 0;
0, y ≤ 0.
DEPARTMENT OF MATHEMATICS & STATISTICS 29
Thus
E
(
1
X
)
= E
( n
Y
)
=
∫ ∞
0
n
y
θnyn−1e−θy
(n − 1)! dy
= nn − 1θ
∫ ∞
0
θn−1yn−2e−θy
(n − 2)! dy
= nn − 1θ,
as long as n ≥ 2.
I So 1X is a biased estimator of θ.
I But n−1nX is unbiased.
I Note that 1X is the MLE of θ.
DEPARTMENT OF MATHEMATICS & STATISTICS 30
Mean Square Error and Variance
I Consider two unbiased estimators, T1 and T2, of τ (θ)
I We say that T1 is more efficient than T2 if
var (T1) < var (T2) ,∀θ ∈ Θ.
I Problem: Not all estimators are unbiased.
Definition: Mean Square Error (MSE)
Let T be an estimator of τ (θ) The Mean Square Error (MSE) of T is
MSE(T , τ(θ)) = E
[
{T − τ (θ)}2
]
which is a function of θ
E
[
{T − τ (θ)}2
]
= E
[
{T − E (T ) + E (T )− τ (θ)}2
]
= E
{
(T − E (T ))2
}
+ {E (T )− τ (θ)}2
+ 2E [(T − E (T )) {E (T )− τ (θ)}]
= var(T ) + (bias(T ))2 + 2 {E (T )− τ (θ)}E (T − E (T ))
= var(T ) + (bias(T ))2 .
DEPARTMENT OF MATHEMATICS & STATISTICS 31
Relative efficiency
I It is better to compare the MSE of the two estimators than the
variance alone.
Definition: Relative Efficiency
Define the efficiency of T1 relative to T2 (in estimating τ (θ)) as
Eff(T2,T1, τ(θ)) =
MSE (T2, τ(θ))
MSE (T1, τ(θ))
.
I If Eff(T2,T1, τ(θ)) < 1, then T2 is said to be a better (more efficient)
estimator than T1.
I Eff(T2,T1, τ(θ)) depends on θ, and may be greater than 1 for some
values of θ and less than 1 for others.
DEPARTMENT OF MATHEMATICS & STATISTICS 32
Consistency
I A sequence of estimators {Tn} of τ (θ) is said to be (weakly)
consistent if
lim
n→∞P (|Tn − τ (θ)| ≤ ε) = 1
∀θ and ∀ε > 0. In other words, the probability that Tn is arbitrarily
close to τ (θ) converges to 1 as the sample size increases.
I We can often prove weak consistency using Tchebysheff’s inequality.
P (|X − µ| > k) ≤ var(X)k2 .
I Consider X as an estimator of E (Xi) = µ. Recall - E
(
X
)
= µ -
var(X) = σ2n .I Thus
P
(∣∣X − µ∣∣ ≤ ε) ≥ 1− var(X)
ε2
= 1− σ
2
nε2 → 1
as n increases, for any fixed ε > 0.
I Hence X is a consistent estimator of µ.
DEPARTMENT OF MATHEMATICS & STATISTICS 33
Properties of MLEs I
1. Invariance: If h is any function, then the MLE of h (θ) is h(θ̂), where
θ̂ is the MLE of θ. Note that h does not have to be one-to-one.
2. Asymptotic Normality: Often the MLE satisfies a central limit
theorem. Under certain conditions, which involve the existence of
certain moments, we have the following result. We assume here that
θ is one-dimensional. There is a more general result when θ has more
than one component. Suppose X1,X2, . . . ,Xn are i.i.d. Let L (θ;X )
be the log-likelihood, θ̂ be the MLE of θ and let
I = −E
{
∂2L (θ;X )
∂θ2
}
.
I is called the information, or Fisher information. Then
P
(√
I
(
θ̂ − θ
)
≤ x
)
→ Φ (x)
= 1√
2pi
∫ x
−∞
e− u
2
2 du,
where Φ is the standard normal cdf.
DEPARTMENT OF MATHEMATICS & STATISTICS 34
Properties of MLEs II
I I may also be calculated using I = E
[{
∂L(θ;X)
∂θ
}2]
.
I We say that the ‘asymptotic variance’ of θ̂ is I−1. We are not saying
that the variance of θ̂ is approximately I−1. The variance of θ̂ may
not even exist for any n.
I For the result to hold, the support of the pdf of X must not depend
on θ. That is, the range of X cannot depend on θ. This rules out, for
example, the case where the Xi are i.i.d. U (0, θ) .
I It is not hard to show, as well, that if τ (θ) is a ‘nice’ function of θ,
then
P
(√J (τ (θ̂)− τ (θ)) ≤ x)→ Φ (x) ,
where
J = I{ d
dθ τ (θ)
}2 .
The asymptotic variance of the MLE τ
(
θ̂
)
of τ (θ) is thus{
d
dθ τ (θ)
}2
I−1.
DEPARTMENT OF MATHEMATICS & STATISTICS 35
Asymptotic MLE Examples
1. {Xi} i.i.d. and exponential with mean 1/θ. Then
fX (x ; θ) =

n∏
i=1
(
θe−θxi
)
; x1, x2, . . . , xn > 0
0 ; otherwise.
=
{
θne−θ
∑n
i=1
xi ; x1, x2, . . . , xn > 0
0 ; otherwise.
L (θ;X ) =
{
θne−θ
∑n
i=1
Xi ; θ > 0
0 ; otherwise.
and
L (θ;X ) = n log θ − θ
n∑
i=1
Xi .
I Thus
∂L (θ;X)
∂θ
= n
θ
−
n∑
i=1
Xi
and
∂2L (θ;X)
∂θ2
= − n
θ2
.
DEPARTMENT OF MATHEMATICS & STATISTICS 36
I Hence
I = n
θ2
and it follows that √n
(
θ̂ − θ
)
/θ is asymptotically N (0, 1) . Note
that the expectation of the second derivative is easier to calculate in
this case than the expectation of the square of the first derivative
(which we don’t even bother calculating).
2. {Xi} i.i.d. Poisson(θ) . Then
fX (x ; θ) =
n∏
i=1
(
e−θθxi
xi !
)
= e−nθ θ
∑n
i=1
xi
n∏
i=1
xi !
,
and so
L (θ;X ) = −nθ +
n∑
i=1
Xi log θ −
n∑
i=1
log (Xi !) .
DEPARTMENT OF MATHEMATICS & STATISTICS 37
I Thus
∂L (θ;X )
∂θ
= −n +
∑n
i=1 Xi
θ
and
∂2L (θ;X )
∂θ2
= −
∑n
i=1 Xi
θ2
.
I We need to calculate I. Each of the techniques involves a bit of work.
However, there is another magic formula! It turns out that
E
{
∂L (θ;X )
∂θ
}
= 0
under very weak conditions (i.e. for just about any pdf or pf).
I Hence in this case
E
( n∑
i=1
Xi
)
= nθ
and so
I = nθ
θ2
= n
θ
.
I Consequently, √n
(
θ̂ − θ
)
/
√
θ is asymptotically N (0, 1) .
DEPARTMENT OF MATHEMATICS & STATISTICS 38
Bootstrap method (nonparametric)
I Standard parametric estimation considers a random sample X with
elements Xi i.i.d.∼ fX (·; θ) for a known density fX that depends on θ.
I Data is observed x =
(
x1 x2 . . . xn
)T
I θ is estimated in various ways. The MLE was used in the previous
section which has nice asymptotic properties.
I Suppose we estimate θ with the sample statistic t(X). How well does
the estimator (and resulting estimates) behave with finite n?
I Bootstrapping (a resampling method) can give an idea of this!
I The nonparamteric resampling using the ECDF.
Definition: Empirical Distribution Function (ECDF)
Let X =
(
X1 X2 . . . Xn
)T be a random sample with each Xi having a
common cumulative distribution function (cdf) F (·). The ECDF is defined,
F̂n(y) =
1
n
n∑
i=1
1{Xi≤y}
DEPARTMENT OF MATHEMATICS & STATISTICS 39
Nonparametric bootstrap idea
I Suppose there is an observed sample of size n,
x =
(
x1 x2 . . . xn
)T
I Create a new sample of size n from the observed data using the
ECDF.
I Take the sample with replacement
I This results in some repeated observations (and some omitted).
I Estimate the parameter θ using the new sample.
I Repeat the sampling and estimation to induce a distribution of θ̂
DEPARTMENT OF MATHEMATICS & STATISTICS 40
More formal Nonparametric bootstrap steps
1. Compute the ECDF, F̂n(·) using x.
2. Draw a sample from F̂n
I Denote this new (bootstrap) sample x i∗ =
(
x1∗ x2∗ . . . xn∗
)T
I Compute a new estimate of θ denoted θ̂1∗ = t(x i∗), using the
bootstrap sample above.
3. Repeat step 2. m times to obtain a set of bootstrap estimates
I (θ̂1∗ θ̂2∗ . . . θ̂m∗)
4. Inspect the distribution of the bootstrap estimates to determine
properties of θ̂.
I Or compute summary properties from the induced bootstrapped
distribution.
DEPARTMENT OF MATHEMATICS & STATISTICS 41
Properties from bootstrapped estimates
I NB: θ̂ = t(x), the estimate of θ from the original sample data.
I Many statistical property measures of θ̂ can be estimated.
I Bootstrap estimate of the mean of θ̂:
Êθ̂ = θ̂∗ =
1
m
m∑
i=1
θ̂i∗
I Leads to estimate of the Bias, B̂ias(θ̂) = Êθ̂ − θ̂
I Bootstrap estimate of the variance of θ̂:
v̂ar(θ̂) = 1m − 1
m∑
i=1
(
θ̂i∗ − θ̂∗
)2
I Bootstrap estimate of the MSE of θ̂.
M̂SE(θ̂, θ) = 1m
m∑
i=1
(
θ̂i∗ − θ̂
)2
DEPARTMENT OF MATHEMATICS & STATISTICS 42
Example: Nonparametric bootstrap
I Suppose Xi − 5 i.i.d.∼ t1,
I Heavy tailed t1 distribution centered at µ = 5.
I Population median and mean are M = µ = 5.
I Simulate a sample of 9 individuals.
n = 9; mu = 5; sig = 1; nu = 1;
set.seed(8)
x = round(rt(n, df = nu) + mu, 2)
I x =
(
4.84 4.28 4.55 5.21 6.02 4.8 6.35 5.01 0.42
)T
I Sample median x˜ = x(5) = 4.84 and mean x = 4.6088889
I Leads to initial estimates
I µ̂1 = x˜ = 4.84
I µ̂2 = x = 4.6088889
DEPARTMENT OF MATHEMATICS & STATISTICS 43
Resample m = 1000 samples
I We could compute the ECDF and sample from there.
I However, easier to use the sample function in R.
m = 1000
Boot = sapply(1:m, function(i) sample(x, replace = TRUE))
Boot
# [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
# [1,] 5.01 6.35 4.84 0.42 0.42 5.21 5.01 6.02 6.35 5.21 6.02 5.21 0.42
# [2,] 6.02 6.02 4.28 5.21 6.02 4.84 4.80 4.55 5.21 4.28 4.28 6.35 4.84
# [3,] 4.80 6.02 4.55 6.02 4.28 4.55 5.21 4.28 6.35 5.01 5.21 6.35 4.84
# [4,] 4.55 4.84 0.42 5.21 0.42 6.02 4.55 4.80 6.02 4.28 0.42 4.55 6.35
# [5,] 0.42 5.21 4.28 6.02 4.28 4.84 6.35 6.02 6.02 6.02 4.55 6.02 6.35
# [6,] 5.01 6.35 5.21 4.80 4.80 4.55 4.80 0.42 4.28 4.84 0.42 6.02 4.84
# [7,] 5.21 4.55 0.42 4.28 5.21 4.28 4.80 4.80 4.28 4.28 4.84 4.55 4.80
# [8,] 4.55 4.28 0.42 5.21 0.42 5.21 6.35 5.21 4.84 5.01 5.01 6.02 4.55
# [9,] 4.80 4.28 4.80 4.28 4.84 0.42 6.02 6.35 4.80 5.01 6.35 4.80 4.80
# [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24]
# [1,] 5.21 0.42 5.21 4.55 4.55 4.80 4.28 6.02 4.84 6.35 6.02
# [2,] 5.21 5.21 6.02 5.01 4.80 6.35 4.84 4.84 4.55 5.01 5.01
# [3,] 4.28 0.42 5.21 4.80 4.84 4.28 4.80 0.42 6.02 4.55 4.80
# [4,] 4.84 4.28 4.28 4.84 4.84 6.02 4.28 6.35 4.55 5.21 0.42
# [5,] 4.80 4.28 5.01 0.42 4.28 6.02 5.01 0.42 5.21 4.28 5.21
# [6,] 5.01 0.42 4.84 0.42 5.21 4.84 6.02 5.21 6.02 5.01 4.80
# [7,] 0.42 6.02 5.21 5.01 4.28 4.28 5.01 5.21 6.02 6.35 5.01
# [8,] 5.01 4.80 0.42 4.28 4.28 4.80 4.28 4.84 5.21 4.55 0.42
# [9,] 4.55 4.55 4.84 4.28 6.02 4.28 5.21 0.42 6.02 5.21 5.01
# [,25] [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35]
# [1,] 4.28 5.01 0.42 4.28 5.01 5.21 5.01 4.28 6.35 5.21 4.28
# [2,] 5.21 0.42 6.02 4.80 4.84 6.35 4.55 4.84 0.42 4.80 4.55
# [3,] 5.21 5.01 4.28 5.01 4.80 4.55 6.02 4.55 4.28 5.21 6.02
# [4,] 6.02 4.28 4.84 0.42 6.35 6.02 5.21 4.55 4.28 5.01 6.02
# [5,] 5.21 4.28 6.02 5.01 4.84 6.35 4.28 4.84 4.80 4.28 6.35
# [6,] 0.42 5.01 6.35 6.02 4.84 4.28 5.01 4.80 4.28 6.02 0.42
# [7,] 0.42 4.28 4.55 5.21 4.55 4.55 6.02 6.02 6.02 5.21 5.21
# [8,] 6.35 6.35 4.55 0.42 0.42 4.55 0.42 6.35 4.84 6.35 6.35
# [9,] 4.28 4.28 4.55 5.01 0.42 4.28 6.02 4.55 4.84 4.55 4.84
# [,36] [,37] [,38] [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46]
# [1,] 4.84 4.80 5.01 4.28 4.80 4.55 4.28 6.35 4.84 6.35 4.84
# [2,] 4.84 4.84 0.42 6.02 4.55 6.02 4.55 6.35 4.84 5.21 6.35
# [3,] 0.42 6.35 4.28 4.84 4.84 4.80 4.84 4.28 4.55 4.84 5.01
# [4,] 5.21 4.28 6.35 5.01 4.84 0.42 4.55 6.02 4.28 4.55 5.01
# [5,] 6.02 4.84 6.02 0.42 5.21 6.35 4.55 6.02 4.80 5.01 4.84
# [6,] 4.84 4.80 6.35 5.01 4.80 5.21 4.84 5.01 5.21 4.84 5.01
# [7,] 0.42 5.01 6.02 4.84 4.55 5.01 4.84 5.21 5.01 6.35 4.84
# [8,] 5.21 5.21 4.28 6.02 6.35 4.28 5.01 5.01 4.84 4.84 4.55
# [9,] 4.84 5.01 5.21 0.42 4.84 4.80 4.55 5.21 4.55 4.28 4.28
# [,47] [,48] [,49] [,50] [,51] [,52] [,53] [,54] [,55] [,56] [,57]
# [1,] 4.80 5.21 5.01 6.02 5.21 6.35 4.28 6.35 4.84 4.28 6.35
# [2,] 6.02 5.21 4.84 4.55 5.21 6.02 6.02 4.55 4.28 6.35 4.28
# [3,] 0.42 4.28 4.28 6.02 6.35 6.35 6.35 6.02 5.01 4.80 4.28
# [4,] 4.80 5.21 4.55 6.35 6.35 5.21 4.80 6.35 0.42 4.80 5.01
# [5,] 4.55 0.42 6.35 4.55 4.84 0.42 6.35 0.42 5.21 4.80 0.42
# [6,] 5.21 6.35 5.01 4.80 4.28 0.42 4.28 6.35 6.35 4.55 5.21
# [7,] 4.84 4.84 0.42 6.35 5.21 4.84 5.21 4.80 4.55 4.55 6.35
# [8,] 6.02 5.21 6.02 0.42 5.21 4.84 4.84 5.21 6.02 5.21 4.55
# [9,] 5.21 4.28 5.21 5.21 4.84 6.02 4.28 4.80 4.28 5.01 4.84
# [,58] [,59] [,60] [,61] [,62] [,63] [,64] [,65] [,66] [,67] [,68]
# [1,] 4.80 4.55 4.55 0.42 4.28 4.55 6.02 6.35 4.28 6.35 4.28
# [2,] 4.28 5.21 0.42 5.01 4.84 4.80 4.80 4.28 0.42 0.42 6.35
# [3,] 6.02 5.01 4.80 6.35 6.02 6.02 0.42 0.42 4.55 6.02 6.02
# [4,] 5.01 0.42 6.02 4.55 4.84 6.35 4.28 5.01 4.80 6.35 6.35
# [5,] 4.84 4.28 4.28 4.28 4.55 6.35 4.28 5.01 5.21 4.80 4.84
# [6,] 5.21 0.42 6.35 0.42 6.35 5.21 4.55 4.28 4.80 6.35 5.21
# [7,] 5.21 6.35 4.28 5.01 5.01 5.21 6.35 4.28 4.28 4.80 6.02
# [8,] 6.35 6.02 6.35 6.35 0.42 6.35 4.28 6.02 5.21 0.42 6.35
# [9,] 4.84 4.55 4.28 4.80 4.28 4.80 4.28 5.01 4.80 6.35 5.01
# [,69] [,70] [,71] [,72] [,73] [,74] [,75] [,76] [,77] [,78] [,79]
# [1,] 6.02 4.28 0.42 6.35 6.35 4.55 4.84 6.02 4.84 4.84 0.42
# [2,] 6.02 6.35 6.02 6.02 4.80 4.28 6.35 4.80 6.35 4.55 0.42
# [3,] 4.28 0.42 4.55 6.35 0.42 5.21 4.28 4.80 6.35 6.02 6.02
# [4,] 4.28 4.84 4.28 6.02 4.28 4.55 4.80 4.84 4.84 4.28 4.80
# [5,] 6.35 4.80 4.55 5.01 4.80 4.80 4.84 4.80 4.84 0.42 4.28
# [6,] 4.55 0.42 6.35 4.55 4.28 6.02 4.84 6.02 4.28 4.84 6.35
# [7,] 4.84 4.80 5.21 4.55 4.55 4.84 4.80 4.55 4.55 4.80 4.28
# [8,] 4.28 4.80 4.28 0.42 0.42 5.01 4.55 5.21 4.84 4.28 6.35
# [9,] 4.28 4.28 6.35 5.01 6.35 6.02 4.55 4.80 5.21 4.84 5.01
# [,80] [,81] [,82] [,83] [,84] [,85] [,86] [,87] [,88] [,89] [,90]
# [1,] 0.42 4.28 6.35 6.35 6.02 4.55 6.02 4.28 5.01 0.42 6.02
# [2,] 5.01 5.21 6.02 6.35 4.80 5.21 4.28 6.02 5.21 5.21 6.02
# [3,] 4.55 5.21 5.01 4.84 0.42 4.55 4.28 5.21 4.55 6.35 4.28
# [4,] 4.80 6.35 4.84 5.01 6.02 6.02 4.84 6.35 0.42 5.01 4.80
# [5,] 4.55 5.01 4.80 5.21 4.84 6.02 4.55 5.01 4.55 5.21 4.28
# [6,] 5.21 4.84 0.42 4.80 4.84 4.80 4.80 4.84 4.28 4.80 4.80
# [7,] 4.55 6.02 0.42 4.55 5.01 6.02 4.84 5.01 0.42 4.84 4.55
# [8,] 6.35 4.84 4.55 6.02 4.28 4.28 5.21 5.01 6.02 6.35 4.28
# [9,] 4.84 4.84 5.21 4.84 5.01 5.21 4.84 6.35 4.80 4.28 5.21
# [,91] [,92] [,93] [,94] [,95] [,96] [,97] [,98] [,99] [,100] [,101]
# [1,] 4.55 6.02 5.01 4.80 5.21 4.80 5.01 5.01 4.84 6.35 6.35
# [2,] 4.55 0.42 6.35 4.28 4.28 4.80 5.21 4.28 6.35 6.35 0.42
# [3,] 4.55 4.28 5.01 6.35 4.84 4.84 4.84 4.84 4.28 4.80 4.84
# [4,] 6.35 4.55 5.01 6.02 4.84 6.35 6.02 4.84 6.35 6.35 0.42
# [5,] 0.42 0.42 0.42 6.02 4.84 0.42 0.42 6.02 4.55 4.28 0.42
# [6,] 4.55 0.42 6.02 4.55 6.35 4.28 4.80 4.84 6.35 4.28 4.55
# [7,] 4.80 6.35 6.35 4.80 4.55 4.28 4.84 5.01 5.21 4.80 4.80
# [8,] 5.21 4.84 0.42 4.84 6.02 5.21 6.35 6.02 6.02 4.55 4.28
# [9,] 5.21 5.01 5.21 4.55 4.84 6.35 5.21 6.35 5.21 4.55 5.01
# [,102] [,103] [,104] [,105] [,106] [,107] [,108] [,109] [,110]
# [1,] 4.55 4.55 4.84 0.42 6.35 0.42 5.01 6.02 6.35
# [2,] 0.42 4.55 4.84 0.42 6.35 0.42 5.01 5.21 4.28
# [3,] 4.80 0.42 6.35 4.84 5.01 4.55 4.84 5.01 4.55
# [4,] 4.28 4.80 6.35 4.80 4.28 5.01 0.42 0.42 4.84
# [5,] 5.21 5.01 5.01 0.42 5.01 0.42 6.35 0.42 5.01
# [6,] 6.35 4.84 6.35 5.21 4.55 4.84 4.80 6.02 4.84
# [7,] 6.02 4.84 0.42 5.21 0.42 5.01 5.21 4.28 4.80
# [8,] 4.80 4.28 0.42 6.35 4.28 0.42 4.55 4.84 4.80
# [9,] 0.42 0.42 5.01 4.28 6.02 5.21 4.80 5.01 6.02
# [,111] [,112] [,113] [,114] [,115] [,116] [,117] [,118] [,119]
# [1,] 6.35 4.80 4.28 4.84 0.42 6.35 4.55 4.80 6.35
# [2,] 6.35 4.80 6.35 6.35 6.35 5.01 4.84 4.84 5.01
# [3,] 4.28 6.35 4.84 5.21 4.80 5.21 4.84 5.01 5.01
# [4,] 6.02 6.02 4.28 6.35 4.84 5.01 4.28 0.42 6.02
# [5,] 4.55 4.84 4.55 4.55 4.84 6.02 6.02 0.42 4.84
# [6,] 0.42 0.42 6.35 0.42 4.80 4.84 4.80 4.55 0.42
# [7,] 5.21 5.21 5.21 4.55 6.35 4.80 0.42 4.28 4.28
# [8,] 5.21 5.21 6.35 5.01 4.84 6.35 4.55 4.80 6.35
# [9,] 5.21 6.02 5.21 4.84 4.80 0.42 6.35 4.80 4.80
# [,120] [,121] [,122] [,123] [,124] [,125] [,126] [,127] [,128]
# [1,] 4.84 0.42 0.42 6.35 6.35 0.42 4.84 4.28 0.42
# [2,] 6.02 4.84 0.42 6.35 6.35 5.21 5.21 4.55 0.42
# [3,] 4.80 5.21 6.02 4.28 6.35 6.35 4.84 4.80 5.01
# [4,] 4.55 4.84 4.28 4.55 6.02 6.35 0.42 6.35 5.21
# [5,] 0.42 4.55 0.42 6.02 5.01 6.35 4.84 6.02 4.84
# [6,] 5.01 4.84 4.80 5.01 6.02 5.01 6.02 6.35 5.01
# [7,] 4.80 5.01 4.84 4.84 4.80 4.55 5.21 4.80 6.02
# [8,] 5.21 5.21 4.55 6.35 0.42 6.35 6.35 4.55 4.84
# [9,] 4.28 5.21 5.21 4.55 6.02 0.42 4.55 6.02 5.01
# [,129] [,130] [,131] [,132] [,133] [,134] [,135] [,136] [,137]
# [1,] 4.28 4.55 4.55 5.01 0.42 0.42 6.35 5.21 4.28
# [2,] 4.55 5.21 4.28 6.02 5.01 4.55 6.35 6.02 6.02
# [3,] 0.42 5.01 5.01 4.55 4.28 6.02 4.80 5.01 4.55
# [4,] 4.80 4.28 6.02 5.01 4.84 0.42 0.42 4.80 4.80
# [5,] 0.42 5.21 5.21 4.80 4.84 4.84 4.28 4.80 4.84
# [6,] 4.80 4.84 0.42 4.55 4.55 6.02 4.84 6.02 0.42
# [7,] 6.02 4.55 4.28 4.55 5.21 0.42 4.80 4.55 4.55
# [8,] 6.35 0.42 4.80 6.02 5.21 4.84 4.80 5.21 4.80
# [9,] 0.42 6.02 6.35 6.02 4.28 0.42 4.55 4.84 4.80
# [,138] [,139] [,140] [,141] [,142] [,143] [,144] [,145] [,146]
# [1,] 4.28 6.35 4.55 5.21 6.02 5.01 0.42 4.55 0.42
# [2,] 4.84 4.80 4.55 4.28 4.84 4.28 5.21 4.55 4.84
# [3,] 6.35 0.42 5.21 5.21 5.21 4.84 6.02 5.21 5.01
# [4,] 6.35 0.42 6.02 4.84 4.55 6.35 6.02 6.02 4.55
# [5,] 6.02 4.55 4.55 5.01 4.28 5.21 5.01 6.02 4.84
# [6,] 6.02 4.84 6.35 5.21 4.28 6.02 6.35 6.35 4.28
# [7,] 0.42 4.84 6.02 5.21 5.01 0.42 4.55 4.84 5.01
# [8,] 6.35 5.01 0.42 4.28 4.80 5.21 4.80 6.02 5.21
# [9,] 4.55 4.55 6.35 4.28 4.84 6.35 5.21 6.35 6.35
# [,147] [,148] [,149] [,150] [,151] [,152] [,153] [,154] [,155]
# [1,] 0.42 6.02 6.35 4.55 4.55 4.28 0.42 4.28 4.55
# [2,] 6.02 0.42 4.84 4.84 4.55 4.80 6.35 6.02 5.01
# [3,] 5.21 5.21 6.02 5.21 4.84 0.42 5.01 5.01 5.01
# [4,] 4.84 0.42 0.42 4.55 6.35 5.01 4.80 4.80 5.01
# [5,] 4.28 4.28 4.55 6.02 4.55 4.28 6.02 4.84 5.21
# [6,] 4.80 5.01 6.35 6.02 4.84 4.55 4.28 4.28 6.35
# [7,] 6.35 5.21 5.21 4.80 5.21 4.28 4.28 4.84 4.55
# [8,] 6.35 4.80 6.35 5.01 4.84 4.28 6.35 5.21 0.42
# [9,] 0.42 4.28 4.84 5.21 4.80 6.02 5.01 6.35 5.01
# [,156] [,157] [,158] [,159] [,160] [,161] [,162] [,163] [,164]
# [1,] 5.01 4.84 5.01 4.28 6.02 6.02 0.42 4.55 0.42
# [2,] 5.01 4.55 6.02 5.21 4.55 0.42 6.35 4.84 4.28
# [3,] 4.84 4.55 0.42 6.02 5.01 6.02 5.21 5.21 4.80
# [4,] 5.01 4.28 4.84 0.42 5.01 6.35 4.55 4.28 6.35
# [5,] 5.01 5.01 4.55 5.01 4.28 4.28 4.55 4.55 4.28
# [6,] 5.21 5.01 4.84 5.01 5.21 4.28 5.21 6.35 4.55
# [7,] 6.02 5.21 6.02 4.84 4.84 4.80 6.02 0.42 5.01
# [8,] 5.21 4.55 5.01 5.21 4.84 6.35 4.80 5.21 0.42
# [9,] 4.55 6.35 5.21 4.80 5.21 5.21 4.84 5.01 4.55
# [,165] [,166] [,167] [,168] [,169] [,170] [,171] [,172] [,173]
# [1,] 4.28 4.55 6.35 4.84 4.84 4.28 0.42 4.80 0.42
# [2,] 5.21 5.01 6.02 6.35 5.01 6.02 4.84 0.42 5.21
# [3,] 5.21 4.80 4.55 6.02 5.21 4.84 0.42 5.01 4.80
# [4,] 0.42 4.80 4.80 6.02 6.02 4.80 6.02 4.84 4.28
# [5,] 0.42 0.42 4.55 0.42 4.28 6.35 5.01 5.01 4.55
# [6,] 4.84 5.01 4.28 4.80 0.42 5.21 4.55 0.42 4.80
# [7,] 5.21 6.02 5.21 0.42 4.55 5.21 4.28 4.80 0.42
# [8,] 5.01 6.35 4.80 4.55 0.42 4.84 0.42 4.84 6.02
# [9,] 4.28 5.21 4.84 0.42 5.01 4.55 5.21 5.21 5.01
# [,174] [,175] [,176] [,177] [,178] [,179] [,180] [,181] [,182]
# [1,] 4.80 0.42 4.84 4.28 0.42 6.35 4.28 5.21 5.01
# [2,] 4.28 5.21 0.42 4.55 4.55 6.02 4.55 6.35 6.02
# [3,] 4.84 4.84 5.01 6.02 5.01 0.42 6.02 5.21 6.02
# [4,] 4.28 0.42 4.80 6.35 4.84 4.84 4.55 5.01 4.80
# [5,] 4.28 5.01 0.42 0.42 4.84 0.42 6.35 4.80 4.55
# [6,] 4.28 0.42 5.01 4.80 4.80 6.35 6.02 5.01 6.02
# [7,] 0.42 6.35 0.42 6.02 0.42 4.55 0.42 5.21 0.42
# [8,] 4.84 4.28 4.55 6.35 4.84 4.55 6.35 6.35 4.80
# [9,] 5.21 4.84 0.42 4.80 0.42 4.84 4.80 5.21 6.35
# [,183] [,184] [,185] [,186] [,187] [,188] [,189] [,190] [,191]
# [1,] 4.84 4.28 6.35 5.01 4.55 4.28 4.55 4.80 4.80
# [2,] 5.21 4.55 5.01 4.28 6.35 4.84 4.28 4.84 4.28
# [3,] 4.55 6.02 0.42 4.84 4.80 4.55 4.55 5.21 4.84
# [4,] 6.35 6.35 4.80 4.28 0.42 4.28 5.21 6.35 4.55
# [5,] 4.55 6.02 6.35 5.21 4.80 4.80 0.42 0.42 0.42
# [6,] 5.21 5.21 4.28 4.80 4.84 4.80 5.01 0.42 5.01
# [7,] 4.55 5.21 4.80 4.55 4.84 5.21 4.84 5.21 5.21
# [8,] 4.55 6.35 0.42 4.28 4.55 5.01 5.21 5.01 4.55
# [9,] 5.21 4.80 4.28 4.80 4.80 4.84 5.01 4.84 5.01
# [,192] [,193] [,194] [,195] [,196] [,197] [,198] [,199] [,200]
# [1,] 0.42 4.80 6.02 4.80 6.02 4.80 4.55 6.02 6.02
# [2,] 0.42 4.55 5.21 5.21 6.02 0.42 5.21 4.28 4.28
# [3,] 6.35 4.55 5.21 0.42 5.21 4.80 5.01 4.84 5.01
# [4,] 4.84 0.42 5.21 4.28 4.80 6.02 4.28 4.84 4.80
# [5,] 6.02 4.55 4.80 4.55 4.55 4.84 4.28 4.55 6.02
# [6,] 4.80 5.21 4.84 6.35 4.55 4.84 6.35 4.55 0.42
# [7,] 4.55 4.84 4.84 4.55 5.01 6.02 6.35 4.84 4.55
# [8,] 5.01 0.42 4.55 6.35 4.55 6.02 4.80 4.80 6.35
# [9,] 4.28 4.55 4.55 6.35 5.21 5.01 4.80 5.21 6.35
# [,201] [,202] [,203] [,204] [,205] [,206] [,207] [,208] [,209]
# [1,] 5.01 5.01 4.80 5.01 4.28 5.01 0.42 4.84 5.21
# [2,] 4.80 4.84 4.84 4.28 6.02 6.35 0.42 6.02 5.01
# [3,] 6.02 5.21 4.80 5.01 4.55 4.28 5.01 4.28 6.35
# [4,] 0.42 4.84 5.01 4.80 4.80 4.84 0.42 4.84 4.84
# [5,] 4.55 0.42 4.28 4.84 6.35 4.28 4.84 6.02 6.02
# [6,] 4.80 4.55 4.84 0.42 6.02 5.21 4.84 6.02 4.28
# [7,] 4.80 5.21 5.01 4.55 6.35 4.55 5.01 5.01 4.55
# [8,] 4.84 6.35 0.42 4.84 6.02 6.35 6.02 5.21 5.01
# [9,] 5.21 6.02 5.21 5.21 4.55 5.21 6.35 0.42 5.01
# [,210] [,211] [,212] [,213] [,214] [,215] [,216] [,217] [,218]
# [1,] 4.28 4.80 6.02 4.80 4.28 5.21 6.35 6.35 4.84
# [2,] 5.01 4.80 6.35 4.80 4.55 6.35 4.55 4.55 5.21
# [3,] 4.84 5.21 4.80 4.28 4.55 6.35 4.80 5.01 4.84
# [4,] 4.28 0.42 0.42 4.55 4.28 4.55 5.01 5.01 4.55
# [5,] 4.84 6.02 5.21 0.42 4.55 6.02 4.55 4.80 0.42
# [6,] 4.28 5.21 4.84 0.42 4.84 6.35 6.02 0.42 4.28
# [7,] 5.01 6.02 6.35 0.42 6.35 4.28 4.55 6.35 4.28
# [8,] 4.55 4.28 4.55 6.02 6.02 4.84 4.28 4.80 4.28
# [9,] 4.84 4.84 6.35 4.80 5.21 0.42 4.55 5.21 6.02
# [,219] [,220] [,221] [,222] [,223] [,224] [,225] [,226] [,227]
# [1,] 6.35 6.35 5.01 4.84 5.01 4.28 0.42 5.01 4.84
# [2,] 4.84 4.80 6.35 4.28 5.01 4.80 6.02 6.02 4.28
# [3,] 6.35 6.35 4.28 4.55 4.28 0.42 4.80 5.21 0.42
# [4,] 4.80 0.42 4.28 5.21 5.01 4.80 0.42 6.35 5.01
# [5,] 4.80 6.35 5.01 4.28 0.42 4.55 4.80 5.01 0.42
# [6,] 5.21 5.01 5.01 6.02 6.02 6.02 6.02 5.21 5.21
# [7,] 4.80 0.42 4.28 0.42 6.35 0.42 6.35 6.35 5.01
# [8,] 5.01 4.55 0.42 4.28 5.21 6.02 5.01 5.21 4.28
# [9,] 6.02 0.42 6.02 4.28 0.42 4.28 0.42 4.28 4.55
# [,228] [,229] [,230] [,231] [,232] [,233] [,234] [,235] [,236]
# [1,] 5.21 5.01 4.80 6.02 4.80 4.84 4.28 4.28 4.84
# [2,] 4.80 5.01 5.01 5.01 6.35 4.28 4.80 0.42 5.01
# [3,] 6.35 5.01 0.42 4.80 4.80 4.28 6.02 0.42 6.35
# [4,] 6.02 5.01 0.42 4.55 4.55 6.02 0.42 5.21 4.80
# [5,] 0.42 4.80 4.28 4.84 6.35 4.80 0.42 6.02 6.02
# [6,] 4.80 5.21 6.35 4.55 0.42 4.28 4.55 4.80 4.84
# [7,] 4.80 5.01 5.21 6.02 5.21 6.02 4.55 6.02 6.02
# [8,] 4.84 4.80 4.55 0.42 4.28 4.55 4.80 4.80 4.28
# [9,] 4.28 4.80 0.42 4.84 4.84 5.21 4.84 5.21 5.01
# [,237] [,238] [,239] [,240] [,241] [,242] [,243] [,244] [,245]
# [1,] 6.02 0.42 5.21 4.84 4.55 0.42 6.35 6.35 4.28
# [2,] 4.80 6.35 4.55 5.21 0.42 4.55 4.84 4.55 4.80
# [3,] 5.01 6.02 6.35 6.35 6.35 5.01 6.02 4.80 6.35
# [4,] 4.80 6.35 4.28 5.21 6.35 4.55 4.80 6.35 4.55
# [5,] 4.84 0.42 0.42 4.55 5.21 4.55 0.42 6.02 5.01
# [6,] 6.02 6.35 5.01 0.42 6.02 6.02 0.42 6.02 4.55
# [7,] 5.21 5.01 4.80 6.35 6.35 4.28 6.02 4.28 5.21
# [8,] 6.35 4.55 4.80 4.28 5.01 4.84 4.84 4.84 4.55
# [9,] 4.28 5.01 5.01 5.21 5.01 0.42 0.42 6.35 6.35
# [,246] [,247] [,248] [,249] [,250] [,251] [,252] [,253] [,254]
# [1,] 5.01 5.01 5.01 0.42 4.84 0.42 6.02 4.80 6.35
# [2,] 4.84 6.02 6.02 5.01 6.02 5.21 5.01 5.21 5.01
# [3,] 4.28 4.55 4.84 0.42 4.84 4.55 4.84 4.55 5.01
# [4,] 5.21 5.01 4.80 6.35 4.28 6.02 6.35 6.35 4.28
# [5,] 0.42 5.21 4.84 4.84 4.55 4.28 4.80 0.42 0.42
# [6,] 4.84 4.28 0.42 4.55 5.01 0.42 5.21 6.02 0.42
# [7,] 4.80 4.28 4.55 6.02 5.01 4.28 5.21 4.80 4.55
# [8,] 5.01 4.55 6.02 6.02 4.28 6.35 6.02 5.21 4.55
# [9,] 4.84 4.84 4.55 4.28 6.35 4.80 6.35 5.21 5.01
# [,255] [,256] [,257] [,258] [,259] [,260] [,261] [,262] [,263]
# [1,] 6.02 6.35 4.80 5.21 0.42 5.01 6.02 6.02 4.80
# [2,] 4.84 4.80 4.80 4.55 0.42 4.55 0.42 4.55 4.80
# [3,] 6.02 4.55 4.28 4.84 4.28 5.21 4.55 4.55 5.01
# [4,] 4.55 4.28 4.84 5.01 4.80 6.02 4.80 4.80 6.35
# [5,] 5.01 0.42 6.02 6.35 6.02 6.35 6.35 4.84 4.80
# [6,] 4.84 4.84 4.80 5.01 5.01 4.28 6.02 5.21 4.55
# [7,] 4.80 6.02 4.28 6.02 5.01 0.42 6.35 5.21 4.84
# [8,] 5.01 5.21 6.35 4.28 4.80 4.28 0.42 5.21 6.02
# [9,] 5.21 6.35 5.21 6.35 6.02 4.28 5.21 4.28 4.55
# [,264] [,265] [,266] [,267] [,268] [,269] [,270] [,271] [,272]
# [1,] 6.02 4.84 0.42 4.55 5.21 6.35 6.02 4.84 5.21
# [2,] 4.28 4.55 4.84 4.28 5.21 0.42 4.55 4.84 6.02
# [3,] 6.35 6.02 4.80 4.55 0.42 4.80 4.28 5.21 6.35
# [4,] 6.02 5.21 0.42 4.84 6.35 4.80 5.01 4.55 6.35
# [5,] 6.35 4.55 6.35 4.84 5.21 0.42 5.21 6.02 0.42
# [6,] 0.42 4.55 4.80 4.80 6.02 5.01 4.55 5.01 4.28
# [7,] 6.02 0.42 5.21 5.01 5.21 4.28 4.80 4.55 4.80
# [8,] 5.01 0.42 4.84 6.02 0.42 4.28 4.84 5.21 0.42
# [9,] 4.80 4.55 6.02 4.84 6.35 5.01 5.21 6.02 5.21
# [,273] [,274] [,275] [,276] [,277] [,278] [,279] [,280] [,281]
# [1,] 6.35 4.84 6.02 6.02 4.28 6.35 5.21 5.01 0.42
# [2,] 4.80 4.28 6.02 4.55 4.80 5.21 5.01 5.01 4.55
# [3,] 4.84 0.42 0.42 5.21 0.42 0.42 5.01 4.80 0.42
# [4,] 4.80 6.35 5.21 6.02 4.55 4.80 6.02 6.35 0.42
# [5,] 6.02 5.01 6.35 5.21 4.55 0.42 5.21 5.21 5.21
# [6,] 0.42 0.42 4.80 6.35 0.42 4.28 4.55 5.21 4.80
# [7,] 4.84 4.55 6.02 4.55 4.28 6.35 6.35 6.35 4.80
# [8,] 6.35 4.28 4.84 5.01 4.55 4.84 4.80 6.02 0.42
# [9,] 5.21 4.55 5.01 6.35 5.01 4.80 0.42 0.42 5.21
# [,282] [,283] [,284] [,285] [,286] [,287] [,288] [,289] [,290]
# [1,] 4.28 6.35 0.42 6.35 5.21 4.84 5.21 6.35 4.55
# [2,] 4.80 6.02 4.80 6.35 5.21 6.02 5.21 4.28 4.55
# [3,] 5.01 5.21 4.55 4.84 6.35 4.80 6.35 4.80 4.28
# [4,] 5.01 4.84 4.84 6.35 5.21 4.55 4.28 4.84 4.84
# [5,] 5.01 4.55 0.42 4.84 6.35 5.21 0.42 4.84 5.01
# [6,] 0.42 5.21 6.02 5.21 0.42 4.55 4.55 5.01 5.21
# [7,] 4.28 0.42 4.80 5.21 5.21 5.01 4.28 4.55 4.84
# [8,] 4.84 4.55 6.02 4.28 4.28 4.84 6.35 4.28 6.02
# [9,] 4.80 5.21 5.01 4.84 5.01 4.84 0.42 6.35 4.84
# [,291] [,292] [,293] [,294] [,295] [,296] [,297] [,298] [,299]
# [1,] 0.42 6.35 6.02 6.35 5.21 4.28 4.28 5.01 4.84
# [2,] 4.84 6.02 5.01 4.84 4.80 4.84 6.35 0.42 4.55
# [3,] 5.21 6.35 5.01 5.21 4.80 5.01 4.55 4.55 0.42
# [4,] 6.02 4.28 5.21 6.35 4.28 5.21 0.42 6.35 5.21
# [5,] 5.21 0.42 6.35 0.42 4.80 6.02 5.21 4.28 6.02
# [6,] 5.01 4.55 6.02 4.28 0.42 0.42 4.80 5.01 5.21
# [7,] 4.28 4.28 0.42 5.01 4.55 4.55 6.02 4.84 4.28
# [8,] 0.42 6.02 5.01 4.80 6.35 4.28 4.28 0.42 5.21
# [9,] 0.42 5.01 4.28 5.21 4.84 4.80 5.01 4.84 4.55
# [,300] [,301] [,302] [,303] [,304] [,305] [,306] [,307] [,308]
# [1,] 4.55 4.84 6.02 5.01 6.02 0.42 5.21 5.01 5.21
# [2,] 6.35 4.80 4.80 4.55 4.28 5.01 5.21 4.84 4.55
# [3,] 4.28 6.02 4.80 6.02 4.84 4.80 0.42 0.42 0.42
# [4,] 5.21 4.55 4.55 4.84 4.80 4.84 4.28 4.55 4.80
# [5,] 4.28 5.01 4.80 4.55 5.21 6.35 4.55 4.28 5.01
# [6,] 4.55 4.55 6.02 6.02 6.35 4.55 6.35 0.42 0.42
# [7,] 0.42 4.84 4.28 4.55 4.80 4.84 4.80 6.35 0.42
# [8,] 4.84 5.21 4.84 4.28 5.01 4.84 4.55 5.01 0.42
# [9,] 0.42 5.01 5.21 4.55 4.84 6.02 6.35 6.35 5.21
# [,309] [,310] [,311] [,312] [,313] [,314] [,315] [,316] [,317]
# [1,] 4.55 5.01 4.55 5.01 0.42 6.35 5.21 6.02 4.84
# [2,] 5.01 0.42 4.84 0.42 4.84 5.21 6.35 6.02 5.21
# [3,] 5.21 4.28 6.35 6.35 4.84 5.21 0.42 5.21 0.42
# [4,] 6.35 4.28 4.80 6.35 4.80 4.55 4.80 4.28 4.55
# [5,] 6.02 6.02 0.42 6.35 4.55 6.35 4.84 5.21 5.01
# [6,] 4.80 0.42 6.02 4.84 4.80 5.21 0.42 4.28 6.02
# [7,] 6.02 0.42 4.84 6.35 6.02 0.42 4.28 4.55 5.01
# [8,] 4.55 4.80 5.01 5.21 6.02 5.01 6.35 4.28 6.02
# [9,] 5.01 0.42 6.02 5.21 6.02 4.28 4.28 6.02 5.01
# [,318] [,319] [,320] [,321] [,322] [,323] [,324] [,325] [,326]
# [1,] 4.84 6.35 0.42 4.80 4.28 4.80 5.01 0.42 4.84
# [2,] 4.84 4.84 4.80 4.80 5.21 4.28 5.21 5.01 5.21
# [3,] 4.28 0.42 4.84 4.80 4.55 5.01 6.35 4.84 5.01
# [4,] 5.21 5.21 4.80 4.55 4.84 6.35 5.21 6.02 4.84
# [5,] 4.80 5.21 6.35 6.35 4.80 0.42 6.02 0.42 4.80
# [6,] 6.02 4.28 4.28 6.35 4.55 0.42 4.80 6.02 5.01
# [7,] 4.84 6.02 6.02 6.35 4.84 4.84 4.84 6.02 4.84
# [8,] 5.21 4.84 5.01 6.02 4.28 4.28 6.02 4.80 5.01
# [9,] 4.28 5.01 4.84 5.21 6.02 5.01 4.84 5.21 4.84
# [,327] [,328] [,329] [,330] [,331] [,332] [,333] [,334] [,335]
# [1,] 4.55 6.35 0.42 5.21 4.84 6.02 6.02 0.42 6.02
# [2,] 5.01 5.21 6.02 0.42 4.28 4.80 0.42 4.28 0.42
# [3,] 4.80 4.84 5.01 6.35 5.01 4.55 6.02 4.55 6.02
# [4,] 6.02 4.80 5.21 0.42 4.55 5.01 4.28 4.28 4.55
# [5,] 6.02 4.55 4.55 4.84 4.55 6.02 6.02 6.35 4.84
# [6,] 5.01 4.28 4.80 4.55 5.21 4.80 4.84 0.42 6.35
# [7,] 5.01 0.42 4.28 0.42 4.55 5.01 4.28 6.35 4.55
# [8,] 6.35 0.42 6.02 4.80 5.21 4.28 6.35 5.01 4.80
# [9,] 4.80 4.55 4.55 0.42 0.42 4.80 6.02 6.35 4.80
# [,336] [,337] [,338] [,339] [,340] [,341] [,342] [,343] [,344]
# [1,] 6.35 6.35 5.21 0.42 4.80 4.28 6.35 0.42 4.84
# [2,] 4.80 6.02 6.35 6.02 4.80 4.80 4.84 4.80 0.42
# [3,] 5.01 6.02 4.55 6.02 6.35 0.42 0.42 4.84 5.21
# [4,] 0.42 5.01 0.42 6.02 5.21 4.84 6.02 4.80 5.21
# [5,] 4.55 4.80 0.42 4.84 4.80 5.01 4.28 4.84 5.01
# [6,] 6.02 6.35 5.01 4.28 5.01 4.84 4.28 6.02 5.01
# [7,] 5.21 6.02 4.55 5.21 6.02 4.28 4.80 5.21 4.84
# [8,] 6.35 5.01 4.55 5.01 6.02 4.28 6.02 6.02 5.01
# [9,] 5.01 4.84 5.21 5.21 0.42 5.21 5.21 4.28 5.21
# [,345] [,346] [,347] [,348] [,349] [,350] [,351] [,352] [,353]
# [1,] 6.02 4.55 0.42 4.28 4.80 4.84 5.21 6.02 4.84
# [2,] 5.01 4.55 4.28 6.02 5.21 5.21 6.35 4.80 4.80
# [3,] 4.28 6.35 4.55 4.84 5.01 4.80 4.28 0.42 5.01
# [4,] 6.35 5.01 4.84 6.35 0.42 0.42 6.35 4.84 4.80
# [5,] 6.35 0.42 4.55 4.55 6.35 5.21 0.42 5.01 4.84
# [6,] 6.02 6.02 4.84 5.21 6.35 6.02 5.01 6.02 5.01
# [7,] 6.35 4.55 6.02 0.42 5.21 5.01 4.80 4.84 5.01
# [8,] 4.80 0.42 4.84 4.28 6.02 4.55 0.42 6.02 5.01
# [9,] 6.35 5.21 4.28 6.02 4.84 4.80 6.02 6.02 5.21
# [,354] [,355] [,356] [,357] [,358] [,359] [,360] [,361] [,362]
# [1,] 0.42 5.21 6.02 6.35 5.01 0.42 4.28 4.80 6.02
# [2,] 6.35 6.35 4.84 0.42 5.01 5.21 0.42 4.55 5.01
# [3,] 4.84 5.01 5.21 5.01 6.02 5.21 5.21 4.84 5.01
# [4,] 5.01 4.28 6.35 5.01 5.21 4.80 4.28 4.84 4.28
# [5,] 6.35 4.55 6.35 0.42 4.28 4.55 6.02 5.01 5.01
# [6,] 5.01 5.01 5.21 4.55 5.01 4.80 6.02 4.80 4.55
# [7,] 4.55 4.80 6.35 6.02 4.84 5.01 6.35 6.35 4.55
# [8,] 6.02 4.55 4.55 0.42 4.55 0.42 0.42 0.42 4.84
# [9,] 4.55 5.21 0.42 6.02 4.55 4.84 4.55 5.01 5.21
# [,363] [,364] [,365] [,366] [,367] [,368] [,369] [,370] [,371]
# [1,] 6.02 4.80 5.01 6.02 4.84 6.35 4.80 6.02 0.42
# [2,] 4.84 0.42 5.21 6.35 0.42 6.35 0.42 4.84 4.80
# [3,] 4.80 6.02 6.35 5.01 5.01 4.80 6.02 5.01 5.01
# [4,] 0.42 4.84 4.80 5.21 6.02 4.55 5.21 0.42 4.28
# [5,] 5.01 0.42 6.35 4.80 5.21 6.02 6.35 0.42 0.42
# [6,] 5.01 5.01 5.01 4.28 5.21 4.28 4.80 5.01 0.42
# [7,] 4.80 4.80 4.80 5.01 6.35 4.55 0.42 4.55 5.01
# [8,] 6.35 6.35 4.55 0.42 0.42 0.42 6.35 5.21 0.42
# [9,] 4.28 5.21 4.55 6.35 4.55 5.01 5.01 6.35 4.28
# [,372] [,373] [,374] [,375] [,376] [,377] [,378] [,379] [,380]
# [1,] 4.55 6.35 6.02 4.28 0.42 6.35 5.21 6.02 0.42
# [2,] 0.42 4.80 4.84 5.21 4.84 4.84 6.02 6.02 6.35
# [3,] 4.84 6.35 6.35 6.35 4.28 5.01 4.28 4.55 5.01
# [4,] 5.21 4.28 4.80 6.35 4.55 6.02 5.01 5.01 4.28
# [5,] 0.42 6.35 6.35 4.80 4.84 4.80 5.01 5.01 6.02
# [6,] 0.42 5.01 5.01 4.84 4.80 0.42 4.80 4.28 6.35
# [7,] 5.01 0.42 5.21 4.84 6.35 6.02 4.28 5.21 4.28
# [8,] 6.35 4.80 5.21 6.02 0.42 4.84 6.35 0.42 5.21
# [9,] 4.28 0.42 4.80 6.35 0.42 0.42 4.84 0.42 4.80
# [,381] [,382] [,383] [,384] [,385] [,386] [,387] [,388] [,389]
# [1,] 6.02 5.21 6.35 4.84 6.02 5.21 5.01 4.80 5.01
# [2,] 4.80 4.80 4.55 6.35 6.35 5.01 6.35 5.01 5.21
# [3,] 5.21 4.80 0.42 4.28 4.28 5.01 5.21 0.42 5.21
# [4,] 5.01 4.55 4.80 4.80 5.01 4.28 4.84 6.02 0.42
# [5,] 5.21 4.28 5.01 6.35 6.35 5.01 6.02 4.28 5.01
# [6,] 6.02 4.80 5.21 6.35 5.21 6.02 6.02 4.80 5.21
# [7,] 4.84 5.01 4.80 4.80 6.35 4.55 5.21 0.42 5.21
# [8,] 6.35 6.35 4.84 4.84 4.80 6.35 5.01 6.02 5.01
# [9,] 4.28 5.01 4.84 4.28 4.80 5.01 5.01 6.35 4.84
# [,390] [,391] [,392] [,393] [,394] [,395] [,396] [,397] [,398]
# [1,] 5.01 4.84 6.02 0.42 4.28 0.42 4.28 5.21 5.01
# [2,] 6.02 6.02 4.80 5.21 0.42 5.21 4.55 4.28 4.80
# [3,] 6.02 0.42 4.55 5.01 5.01 4.80 0.42 4.55 6.02
# [4,] 4.55 5.01 6.02 0.42 0.42 6.02 5.01 4.84 0.42
# [5,] 4.55 6.02 4.55 4.80 4.80 4.55 4.84 0.42 6.35
# [6,] 6.02 0.42 4.84 4.84 0.42 5.01 4.28 4.28 0.42
# [7,] 4.55 0.42 6.02 4.84 4.55 5.21 6.35 4.28 4.84
# [8,] 5.01 6.35 5.01 4.28 4.55 0.42 5.21 4.80 5.21
# [9,] 6.35 4.80 4.84 6.02 5.01 4.55 6.02 4.84 5.21
# [,399] [,400] [,401] [,402] [,403] [,404] [,405] [,406] [,407]
# [1,] 6.02 6.35 6.35 4.28 4.28 4.28 5.21 4.55 4.84
# [2,] 5.21 4.28 4.84 0.42 4.28 4.84 0.42 4.80 4.80
# [3,] 4.80 5.21 6.02 5.21 4.55 4.55 4.55 6.35 4.84
# [4,] 6.35 5.01 6.35 5.21 0.42 0.42 6.35 0.42 4.55
# [5,] 6.35 5.01 5.21 4.28 4.84 4.80 6.35 0.42 4.28
# [6,] 5.01 5.01 5.21 6.35 4.28 4.28 6.02 5.21 4.84
# [7,] 0.42 4.28 4.55 5.21 4.80 6.35 0.42 4.80 6.02
# [8,] 4.28 6.02 6.35 6.02 5.21 6.35 4.28 6.35 6.35
# [9,] 4.55 6.35 4.55 4.84 4.80 4.28 0.42 6.35 0.42
# [,408] [,409] [,410] [,411] [,412] [,413] [,414] [,415] [,416]
# [1,] 4.80 0.42 5.01 6.35 4.55 6.02 6.35 5.21 0.42
# [2,] 5.01 0.42 0.42 5.01 6.02 4.28 6.02 5.01 6.35
# [3,] 5.01 5.21 6.35 5.21 4.28 5.21 5.01 4.84 4.28
# [4,] 4.55 4.28 5.21 6.02 4.55 4.55 5.21 0.42 4.55
# [5,] 4.80 4.28 5.01 4.84 4.55 5.21 6.35 4.80 5.21
# [6,] 4.80 4.55 4.80 5.21 4.55 6.35 4.80 4.84 0.42
# [7,] 6.02 4.28 5.21 6.02 6.35 5.21 4.55 0.42 5.01
# [8,] 0.42 0.42 4.84 4.28 6.35 6.35 6.02 4.80 4.80
# [9,] 4.80 4.28 4.80 5.01 4.84 5.01 6.35 5.21 4.80
# [,417] [,418] [,419] [,420] [,421] [,422] [,423] [,424] [,425]
# [1,] 4.80 4.55 5.01 4.55 4.84 4.28 4.80 5.01 5.01
# [2,] 5.01 4.80 4.84 5.01 5.21 4.80 4.84 4.55 5.01
# [3,] 4.80 5.21 6.35 5.01 4.55 6.35 4.28 6.02 5.21
# [4,] 6.02 4.28 4.80 5.21 6.35 4.84 5.01 5.01 4.55
# [5,] 4.80 4.80 5.01 5.01 4.80 4.80 6.35 5.21 4.84
# [6,] 4.80 4.84 4.80 5.01 4.84 4.55 4.55 4.80 4.55
# [7,] 4.84 4.84 4.55 6.02 4.84 6.35 6.35 4.55 4.80
# [8,] 0.42 5.21 4.84 4.55 4.80 6.35 4.80 6.02 4.55
# [9,] 4.55 4.55 5.01 5.21 5.21 6.02 6.02 0.42 4.84
# [,426] [,427] [,428] [,429] [,430] [,431] [,432] [,433] [,434]
# [1,] 4.84 4.28 0.42 4.28 4.84 4.84 5.01 5.21 5.01
# [2,] 4.84 6.35 4.80 4.80 6.35 5.21 4.80 6.35 0.42
# [3,] 5.01 5.21 6.02 4.80 4.80 4.55 0.42 4.84 4.55
# [4,] 4.84 5.01 6.35 5.21 5.01 0.42 6.02 0.42 5.21
# [5,] 5.01 6.35 5.21 5.21 4.28 4.80 4.80 4.55 4.84
# [6,] 6.02 0.42 5.01 4.80 0.42 4.28 4.55 6.02 4.84
# [7,] 4.84 6.02 4.80 5.01 4.28 4.80 4.28 4.55 6.35
# [8,] 5.01 4.84 5.01 6.02 6.02 0.42 5.21 4.55 6.35
# [9,] 6.02 4.55 4.80 0.42 4.28 4.55 6.35 6.35 5.01
# [,435] [,436] [,437] [,438] [,439] [,440] [,441] [,442] [,443]
# [1,] 4.55 6.35 6.35 4.55 4.80 4.55 0.42 6.35 4.55
# [2,] 6.02 4.84 4.80 6.02 5.01 6.02 6.02 5.21 6.35
# [3,] 4.55 4.80 0.42 5.01 0.42 4.84 4.84 5.01 5.21
# [4,] 0.42 4.80 5.21 6.02 6.02 5.21 6.35 4.55 4.55
# [5,] 5.21 4.80 4.28 4.55 5.21 4.28 6.35 6.02 6.02
# [6,] 4.28 5.21 6.35 6.35 4.55 6.35 0.42 5.01 0.42
# [7,] 5.21 6.35 4.84 4.55 5.01 6.02 4.55 4.84 6.02
# [8,] 5.01 4.55 4.28 6.35 4.55 4.84 6.35 4.80 6.35
# [9,] 4.55 5.01 6.35 5.01 4.55 4.80 5.21 6.02 6.35
# [,444] [,445] [,446] [,447] [,448] [,449] [,450] [,451] [,452]
# [1,] 5.01 5.01 5.01 5.21 5.21 4.80 4.80 6.35 0.42
# [2,] 4.80 5.21 4.80 4.55 4.28 6.35 6.02 4.55 5.21
# [3,] 4.80 4.84 4.28 0.42 4.80 4.28 5.21 4.80 6.35
# [4,] 4.55 5.21 6.02 4.84 6.02 4.55 6.02 5.21 6.35
# [5,] 4.28 6.02 4.84 6.35 5.01 5.21 5.01 4.80 4.55
# [6,] 4.55 4.55 5.01 4.28 6.02 4.80 4.80 4.28 6.35
# [7,] 4.80 5.21 5.21 4.28 6.35 6.02 4.28 6.35 5.21
# [8,] 4.55 0.42 5.21 4.55 4.55 4.80 5.21 4.84 4.28
# [9,] 4.80 4.55 5.21 4.55 4.80 4.55 5.21 4.84 5.21
# [,453] [,454] [,455] [,456] [,457] [,458] [,459] [,460] [,461]
# [1,] 4.80 4.84 6.35 4.55 6.02 5.21 4.80 6.35 4.84
# [2,] 4.84 5.01 4.28 0.42 5.01 6.02 6.02 5.21 5.21
# [3,] 4.28 4.55 0.42 4.28 4.84 5.21 4.80 4.80 6.02
# [4,] 4.84 4.28 4.84 5.01 5.01 0.42 0.42 4.84 4.84
# [5,] 4.28 4.80 4.55 5.21 0.42 4.28 6.02 4.28 6.02
# [6,] 6.02 4.55 4.28 6.02 4.80 4.28 0.42 4.55 5.21
# [7,] 4.55 4.80 4.55 5.01 6.02 5.01 4.28 5.01 5.21
# [8,] 5.21 5.21 5.01 5.01 5.21 5.21 5.01 0.42 6.35
# [9,] 4.80 6.35 4.28 5.21 5.21 4.55 4.80 4.28 4.80
# [,462] [,463] [,464] [,465] [,466] [,467] [,468] [,469] [,470]
# [1,] 4.80 4.84 4.80 6.35 0.42 4.84 4.84 0.42 4.28
# [2,] 6.02 4.80 6.35 4.55 4.84 0.42 5.21 4.28 5.01
# [3,] 6.02 6.35 6.02 6.35 5.01 0.42 4.80 6.35 5.01
# [4,] 5.01 0.42 4.80 0.42 6.35 0.42 6.02 4.55 5.21
# [5,] 0.42 0.42 6.35 4.28 5.21 0.42 5.01 6.35 5.01
# [6,] 5.21 5.01 4.28 5.21 4.84 6.02 6.35 0.42 4.28
# [7,] 5.01 4.84 4.55 4.28 5.21 5.21 0.42 6.02 4.28
# [8,] 5.01 5.21 6.02 5.01 4.28 6.35 0.42 6.02 4.55
# [9,] 0.42 6.02 0.42 6.35 4.28 5.21 4.80 4.80 5.21
# [,471] [,472] [,473] [,474] [,475] [,476] [,477] [,478] [,479]
# [1,] 5.21 4.55 5.01 6.02 0.42 5.21 0.42 4.80 5.21
# [2,] 4.84 6.02 4.55 5.01 4.55 5.21 6.35 4.84 4.84
# [3,] 6.35 5.21 6.35 4.28 6.02 5.01 5.01 4.28 0.42
# [4,] 4.55 6.02 4.84 4.84 4.28 5.21 5.01 5.21 4.84
# [5,] 4.28 4.80 5.01 6.35 0.42 4.84 4.55 5.01 4.84
# [6,] 4.84 6.35 4.84 5.01 4.55 6.35 5.01 6.35 4.84
# [7,] 4.28 5.21 4.84 4.55 4.55 4.84 4.80 5.01 6.35
# [8,] 4.84 0.42 5.21 4.84 4.55 5.01 4.28 6.02 5.21
# [9,] 4.80 4.84 5.01 4.84 4.80 4.84 5.21 4.84 5.21
# [,480] [,481] [,482] [,483] [,484] [,485] [,486] [,487] [,488]
# [1,] 4.55 0.42 4.84 4.84 6.02 4.84 5.21 6.35 4.28
# [2,] 4.28 4.84 4.80 4.55 4.84 4.28 4.55 5.21 4.80
# [3,] 4.55 5.01 4.84 4.55 4.28 5.21 4.84 4.28 6.35
# [4,] 6.35 4.84 6.02 5.21 0.42 6.35 6.35 5.21 6.35
# [5,] 5.21 5.21 5.01 6.02 4.28 5.21 4.80 6.02 5.01
# [6,] 4.80 6.02 6.02 4.55 6.35 5.01 4.55 6.35 6.35
# [7,] 6.35 4.55 0.42 5.21 4.55 5.21 5.01 4.84 6.02
# [8,] 5.21 4.28 4.55 6.02 0.42 5.01 0.42 4.80 5.01
# [9,] 4.80 4.80 4.28 6.35 4.28 6.02 6.35 5.21 0.42
# [,489] [,490] [,491] [,492] [,493] [,494] [,495] [,496] [,497]
# [1,] 5.21 4.55 0.42 4.55 0.42 6.02 6.35 5.01 4.84
# [2,] 4.28 6.35 5.01 4.28 0.42 5.21 6.02 4.55 5.01
# [3,] 6.02 4.80 0.42 6.35 6.02 5.21 4.84 6.35 4.55
# [4,] 5.01 5.21 4.55 4.28 4.28 4.80 4.84 4.55 4.80
# [5,] 4.80 4.28 4.55 4.55 4.28 5.01 4.55 5.01 6.02
# [6,] 4.80 6.02 5.21 4.84 5.01 4.55 6.02 4.80 0.42
# [7,] 5.01 4.28 4.80 0.42 4.55 4.55 4.55 5.01 5.21
# [8,] 4.84 4.55 4.55 4.84 4.55 5.21 4.84 4.84 4.55
# [9,] 4.55 4.80 5.01 4.84 6.35 4.84 4.80 6.02 4.80
# [,498] [,499] [,500] [,501] [,502] [,503] [,504] [,505] [,506]
# [1,] 4.55 5.01 4.80 0.42 0.42 4.84 4.84 4.28 5.21
# [2,] 6.02 6.35 5.01 6.35 4.84 4.28 6.35 0.42 6.02
# [3,] 4.84 4.84 4.55 4.28 6.02 6.35 0.42 4.28 6.35
# [4,] 6.35 4.80 0.42 0.42 6.02 4.55 4.80 4.28 5.01
# [5,] 6.35 6.02 4.84 0.42 5.21 4.55 5.01 0.42 4.84
# [6,] 4.80 4.28 4.84 6.02 5.01 5.01 6.02 4.84 0.42
# [7,] 4.55 4.80 5.21 4.80 5.21 4.28 4.84 4.84 5.21
# [8,] 4.84 4.84 0.42 5.21 6.02 6.02 4.55 5.21 4.84
# [9,] 4.28 6.02 6.02 6.02 6.02 4.84 6.35 0.42 4.80
# [,507] [,508] [,509] [,510] [,511] [,512] [,513] [,514] [,515]
# [1,] 5.01 6.35 4.55 5.01 4.28 6.02 5.01 4.84 6.02
# [2,] 6.02 4.55 4.80 0.42 4.55 6.35 6.35 4.55 5.21
# [3,] 6.35 6.35 0.42 4.84 5.01 4.80 4.55 6.02 6.02
# [4,] 6.02 5.21 6.02 4.84 6.35 4.28 5.21 0.42 4.28
# [5,] 4.80 6.02 5.01 0.42 4.28 5.01 5.21 0.42 6.35
# [6,] 4.28 4.28 4.55 5.21 0.42 4.80 4.55 4.80 4.55
# [7,] 5.01 4.84 0.42 4.80 6.02 4.55 5.01 0.42 4.28
# [8,] 6.02 6.35 4.80 6.02 4.28 6.35 5.21 5.01 4.55
# [9,] 4.84 6.35 6.35 0.42 4.28 6.02 4.55 5.01 0.42
# [,516] [,517] [,518] [,519] [,520] [,521] [,522] [,523] [,524]
# [1,] 5.21 5.21 4.55 4.80 4.84 0.42 6.35 6.02 5.01
# [2,] 4.55 6.35 6.02 6.35 6.35 4.84 5.01 0.42 6.35
# [3,] 5.01 4.80 4.80 4.80 4.80 6.02 4.55 5.01 5.01
# [4,] 5.01 4.55 6.02 0.42 6.35 4.80 4.28 4.55 6.02
# [5,] 6.35 5.01 4.55 0.42 0.42 5.01 4.80 4.28 5.21
# [6,] 5.01 6.35 4.55 6.02 5.01 6.35 4.55 6.35 4.55
# [7,] 5.01 4.80 4.28 5.01 4.55 5.01 6.35 5.21 0.42
# [8,] 6.02 4.84 4.80 0.42 6.35 4.55 6.35 5.21 4.28
# [9,] 4.84 5.01 6.02 5.21 5.21 4.55 6.35 4.80 6.35
# [,525] [,526] [,527] [,528] [,529] [,530] [,531] [,532] [,533]
# [1,] 5.21 5.21 4.28 4.55 5.01 6.02 4.55 4.84 4.84
# [2,] 4.28 5.21 6.35 0.42 4.55 5.21 5.01 4.55 5.21
# [3,] 5.21 0.42 0.42 4.84 5.21 4.28 6.02 4.28 4.55
# [4,] 5.01 4.55 4.55 6.02 5.21 6.02 5.01 0.42 4.80
# [5,] 6.02 5.01 0.42 6.35 5.01 5.21 4.55 4.84 6.02
# [6,] 4.80 4.55 4.28 6.02 6.35 5.01 5.21 6.02 4.28
# [7,] 5.01 4.55 6.02 5.21 5.21 0.42 0.42 0.42 5.21
# [8,] 4.80 6.35 6.35 4.28 4.55 6.02 4.55 0.42 5.21
# [9,] 6.02 5.01 4.80 4.84 0.42 0.42 5.21 6.02 5.21
# [,534] [,535] [,536] [,537] [,538] [,539] [,540] [,541] [,542]
# [1,] 4.80 4.84 4.80 4.80 5.01 4.55 5.01 4.55 4.80
# [2,] 4.28 0.42 5.21 5.21 5.01 4.55 6.35 6.35 4.28
# [3,] 6.02 6.35 6.02 0.42 6.35 4.55 4.84 4.84 4.84
# [4,] 4.80 4.28 6.02 4.28 6.02 6.02 4.84 5.21 6.35
# [5,] 4.84 4.84 6.35 6.02 5.21 5.21 6.02 4.84 6.02
# [6,] 6.35 4.55 5.01 4.55 4.80 6.35 6.35 0.42 5.21
# [7,] 4.55 5.21 6.02 4.80 6.35 4.28 4.84 5.21 5.21
# [8,] 4.80 5.21 4.80 6.35 4.55 4.55 5.01 0.42 0.42
# [9,] 5.21 0.42 4.84 4.55 5.01 4.55 4.28 6.35 6.35
# [,543] [,544] [,545] [,546] [,547] [,548] [,549] [,550] [,551]
# [1,] 4.84 4.84 6.02 5.01 4.80 4.28 0.42 4.84 4.55
# [2,] 6.02 4.80 0.42 6.02 4.55 0.42 6.02 5.21 0.42
# [3,] 5.01 6.02 5.21 5.01 4.84 6.02 6.02 5.01 6.02
# [4,] 5.21 5.21 5.21 4.28 4.28 4.84 4.84 6.02 5.01
# [5,] 4.80 4.84 5.01 4.80 6.35 6.02 6.02 4.55 6.02
# [6,] 6.35 4.84 6.02 6.35 6.35 6.35 4.55 6.35 4.84
# [7,] 6.02 4.55 4.80 5.21 4.28 6.02 0.42 0.42 4.28
# [8,] 4.84 6.35 6.35 4.84 0.42 5.21 4.28 4.84 5.01
# [9,] 4.80 6.35 4.55 4.28 0.42 4.55 0.42 4.80 4.28
# [,552] [,553] [,554] [,555] [,556] [,557] [,558] [,559] [,560]
# [1,] 5.21 4.80 4.28 5.01 6.02 6.35 0.42 0.42 6.35
# [2,] 5.21 4.80 4.80 0.42 5.21 4.80 5.01 6.35 4.84
# [3,] 4.28 6.35 5.21 4.84 0.42 6.35 6.35 0.42 5.01
# [4,] 5.01 4.28 5.21 6.35 4.80 4.55 4.55 6.02 4.28
# [5,] 4.28 4.55 0.42 4.28 5.21 6.35 6.02 4.80 5.01
# [6,] 4.80 4.55 0.42 5.01 4.80 6.02 0.42 4.55 5.21
# [7,] 5.01 6.02 5.01 5.21 6.02 4.80 5.21 4.28 6.02
# [8,] 4.80 4.55 6.35 4.80 4.28 4.80 4.84 6.02 4.28
# [9,] 5.21 4.28 4.28 6.35 6.02 4.80 4.28 0.42 4.55
# [,561] [,562] [,563] [,564] [,565] [,566] [,567] [,568] [,569]
# [1,] 4.55 6.35 4.28 6.35 0.42 6.02 5.21 0.42 6.35
# [2,] 4.80 0.42 5.21 5.21 4.55 4.80 5.21 5.01 6.02
# [3,] 4.28 5.21 5.01 6.35 4.84 4.80 6.02 5.01 4.80
# [4,] 4.28 4.55 4.80 4.55 6.35 4.28 6.02 4.84 4.55
# [5,] 4.80 5.01 4.80 4.84 0.42 6.02 0.42 6.02 6.35
# [6,] 4.28 5.01 4.55 6.35 6.02 4.55 6.35 6.02 4.28
# [7,] 4.28 6.35 5.01 4.80 4.84 5.01 4.80 4.80 0.42
# [8,] 0.42 4.84 6.02 6.35 5.21 4.28 5.21 6.35 4.28
# [9,] 6.02 5.21 4.80 6.35 5.01 6.02 5.01 5.21 5.01
# [,570] [,571] [,572] [,573] [,574] [,575] [,576] [,577] [,578]
# [1,] 5.01 6.35 5.21 5.21 0.42 6.35 5.21 6.02 4.80
# [2,] 6.02 0.42 4.55 4.55 5.01 0.42 4.55 5.21 5.21
# [3,] 5.01 4.28 0.42 4.55 6.35 4.28 5.01 6.35 4.55
# [4,] 0.42 6.02 4.80 6.02 4.80 6.02 6.35 0.42 4.55
# [5,] 4.84 6.02 4.80 4.80 4.84 4.80 4.55 6.35 0.42
# [6,] 4.84 4.28 6.02 5.21 5.21 5.01 0.42 6.02 6.35
# [7,] 0.42 5.21 6.02 0.42 4.55 0.42 5.01 0.42 5.01
# [8,] 5.21 4.84 4.28 5.01 0.42 5.21 4.28 5.21 4.84
# [9,] 4.84 5.21 5.01 0.42 6.02 5.21 4.84 4.84 6.02
# [,579] [,580] [,581] [,582] [,583] [,584] [,585] [,586] [,587]
# [1,] 5.01 5.21 4.84 4.28 5.01 6.35 0.42 4.28 4.28
# [2,] 4.80 6.02 0.42 5.01 0.42 5.21 4.28 4.80 0.42
# [3,] 4.80 5.01 4.28 4.80 6.02 4.84 4.55 4.84 4.28
# [4,] 0.42 6.35 6.02 4.28 6.02 6.02 6.02 6.35 6.35
# [5,] 5.01 6.02 5.21 5.01 5.21 4.80 5.21 6.35 4.84
# [6,] 4.28 4.28 4.55 5.01 6.35 5.21 5.01 4.55 4.80
# [7,] 6.35 5.21 4.80 4.55 4.55 4.28 5.21 4.84 4.55
# [8,] 0.42 5.01 0.42 4.80 4.84 4.84 6.35 0.42 4.28
# [9,] 4.80 6.35 4.80 6.35 0.42 0.42 4.84 6.35 4.84
# [,588] [,589] [,590] [,591] [,592] [,593] [,594] [,595] [,596]
# [1,] 5.01 4.84 4.84 4.80 6.02 4.28 6.35 4.55 4.28
# [2,] 4.80 4.55 5.01 4.84 4.84 4.80 6.35 6.02 4.80
# [3,] 4.84 4.80 5.21 4.84 0.42 5.21 5.21 6.02 5.01
# [4,] 5.21 4.80 5.21 4.80 6.35 4.28 0.42 5.01 5.21
# [5,] 6.02 5.21 0.42 5.21 4.80 4.84 5.21 4.84 6.02
# [6,] 4.55 6.35 6.35 0.42 4.80 0.42 4.28 6.35 4.84
# [7,] 5.01 4.28 0.42 4.55 4.80 6.02 4.84 6.35 6.02
# [8,] 5.21 6.35 4.55 6.02 4.84 4.28 4.28 5.21 6.02
# [9,] 4.80 6.02 4.84 6.35 6.02 4.28 4.28 4.28 4.28
# [,597] [,598] [,599] [,600] [,601] [,602] [,603] [,604] [,605]
# [1,] 4.84 5.01 4.28 0.42 4.28 4.80 6.02 4.84 5.21
# [2,] 5.01 4.80 5.21 4.55 4.84 5.21 6.35 5.01 0.42
# [3,] 5.21 6.35 4.55 6.02 5.01 5.01 5.01 4.28 4.80
# [4,] 4.55 4.28 6.35 0.42 4.80 4.55 0.42 6.02 6.35
# [5,] 6.02 4.84 4.80 4.28 4.28 4.80 0.42 6.02 5.01
# [6,] 4.28 4.80 4.84 6.35 4.80 4.55 4.28 6.35 4.80
# [7,] 5.01 4.84 4.55 5.21 6.35 6.35 5.01 4.55 6.35
# [8,] 5.01 4.84 5.01 4.28 6.02 6.02 6.35 5.01 4.80
# [9,] 6.02 6.35 5.21 4.55 4.28 0.42 4.55 4.55 4.55
# [,606] [,607] [,608] [,609] [,610] [,611] [,612] [,613] [,614]
# [1,] 4.80 4.28 6.35 4.84 5.01 5.21 5.01 4.28 4.28
# [2,] 4.28 0.42 5.21 5.21 4.55 5.01 4.55 5.21 4.28
# [3,] 5.21 6.35 0.42 6.02 4.80 5.21 4.55 4.80 4.80
# [4,] 6.35 4.84 5.01 0.42 4.80 5.01 5.01 5.01 4.28
# [5,] 4.28 6.35 6.02 4.84 5.21 6.35 5.21 4.84 0.42
# [6,] 4.28 6.02 0.42 4.84 6.35 4.84 4.55 4.80 5.01
# [7,] 6.35 6.35 0.42 4.55 0.42 6.02 4.28 0.42 6.02
# [8,] 4.84 4.80 4.80 6.02 6.35 5.01 0.42 4.28 4.55
# [9,] 4.28 6.02 4.28 0.42 5.21 0.42 4.80 5.01 0.42
# [,615] [,616] [,617] [,618] [,619] [,620] [,621] [,622] [,623]
# [1,] 6.02 6.35 4.84 4.84 4.80 0.42 4.84 4.84 5.21
# [2,] 6.02 0.42 4.55 4.55 5.01 6.35 4.84 4.80 4.55
# [3,] 0.42 4.55 5.21 5.21 5.01 4.84 5.01 5.01 0.42
# [4,] 4.28 6.35 4.84 0.42 4.80 5.21 4.84 5.21 5.21
# [5,] 6.35 5.21 4.80 6.35 4.55 4.80 4.84 0.42 0.42
# [6,] 5.21 5.01 4.55 6.02 0.42 6.35 4.84 4.80 6.02
# [7,] 4.28 4.80 6.02 4.55 4.84 6.35 4.84 5.01 5.21
# [8,] 5.01 6.02 4.80 4.80 4.28 5.21 6.35 6.35 5.21
# [9,] 6.02 5.01 5.01 4.84 6.35 4.28 6.35 4.55 4.84
# [,624] [,625] [,626] [,627] [,628] [,629] [,630] [,631] [,632]
# [1,] 5.21 5.01 4.84 5.21 4.84 5.01 0.42 6.35 6.35
# [2,] 5.21 4.84 4.55 4.80 6.35 6.35 4.84 6.02 4.28
# [3,] 6.02 6.02 4.84 0.42 5.01 4.55 6.35 4.80 6.35
# [4,] 6.35 5.01 6.35 4.28 4.80 6.35 6.35 0.42 4.28
# [5,] 6.02 5.21 6.02 4.84 6.35 6.35 4.80 6.02 5.21
# [6,] 4.28 4.84 0.42 4.28 5.21 0.42 6.35 5.21 6.35
# [7,] 4.28 4.55 4.55 6.35 6.35 4.28 5.01 4.84 4.28
# [8,] 5.21 6.02 4.55 0.42 4.80 6.02 0.42 6.02 4.28
# [9,] 5.01 5.21 6.02 4.55 4.84 4.84 5.21 4.80 6.02
# [,633] [,634] [,635] [,636] [,637] [,638] [,639] [,640] [,641]
# [1,] 4.84 5.21 4.80 5.01 6.02 5.01 4.55 4.84 5.01
# [2,] 4.55 6.35 6.35 6.35 4.84 4.28 4.84 6.02 4.84
# [3,] 4.84 5.01 4.55 4.80 5.01 0.42 5.21 0.42 6.02
# [4,] 5.21 6.02 4.80 0.42 5.21 4.28 5.21 0.42 6.35
# [5,] 5.21 5.21 4.55 0.42 5.21 6.35 6.02 4.80 4.84
# [6,] 4.55 4.80 6.35 4.84 5.01 4.28 5.01 6.02 4.28
# [7,] 4.55 4.28 0.42 4.84 6.35 4.80 4.80 6.02 4.55
# [8,] 6.35 5.21 5.01 5.21 6.35 4.55 6.35 4.55 5.01
# [9,] 4.80 6.35 4.28 0.42 4.55 4.80 5.01 4.84 4.80
# [,642] [,643] [,644] [,645] [,646] [,647] [,648] [,649] [,650]
# [1,] 6.02 4.55 0.42 0.42 4.84 4.80 4.55 0.42 5.01
# [2,] 4.28 4.28 6.35 4.55 6.35 6.35 4.55 5.01 4.55
# [3,] 6.02 0.42 6.02 6.35 4.28 4.84 5.21 5.01 4.55
# [4,] 4.55 4.84 5.01 6.02 0.42 4.55 6.35 6.02 4.84
# [5,] 6.02 6.35 6.35 0.42 6.35 5.21 4.84 5.01 5.21
# [6,] 0.42 0.42 6.02 0.42 4.84 6.02 4.55 6.35 4.80
# [7,] 6.35 5.21 0.42 5.01 6.35 5.21 4.28 4.28 0.42
# [8,] 6.35 4.55 4.80 5.01 0.42 4.80 4.28 4.55 0.42
# [9,] 4.80 0.42 5.01 4.84 4.28 6.02 4.80 5.21 4.55
# [,651] [,652] [,653] [,654] [,655] [,656] [,657] [,658] [,659]
# [1,] 5.21 0.42 5.01 6.35 5.01 4.80 4.55 4.55 5.21
# [2,] 4.55 4.80 4.28 6.02 4.80 6.35 0.42 0.42 6.02
# [3,] 4.28 5.01 6.35 5.01 5.01 4.28 6.02 4.55 4.55
# [4,] 5.21 6.35 5.01 6.02 5.01 4.80 6.02 4.84 5.21
# [5,] 6.02 5.21 5.21 4.84 5.01 6.35 4.80 4.80 5.01
# [6,] 6.02 0.42 4.55 4.28 5.01 4.55 4.80 5.21 4.28
# [7,] 6.35 4.84 4.55 5.01 0.42 6.02 4.55 4.84 4.55
# [8,] 0.42 6.02 4.28 6.35 4.80 4.80 4.55 6.02 6.02
# [9,] 0.42 6.02 6.02 5.21 4.28 5.21 5.21 5.21 5.01
# [,660] [,661] [,662] [,663] [,664] [,665] [,666] [,667] [,668]
# [1,] 5.01 5.01 0.42 4.80 6.35 5.01 6.02 4.55 6.02
# [2,] 5.21 4.80 5.01 4.84 0.42 4.55 6.35 4.28 4.55
# [3,] 4.55 4.28 5.01 6.02 6.35 4.28 5.21 6.02 4.80
# [4,] 6.35 4.55 6.35 5.01 6.02 0.42 5.21 4.28 0.42
# [5,] 5.21 4.84 4.80 4.80 4.28 4.84 4.28 6.02 5.21
# [6,] 4.55 6.02 6.02 4.28 6.02 4.55 4.80 6.02 0.42
# [7,] 5.21 4.80 6.02 6.02 0.42 0.42 5.21 0.42 5.21
# [8,] 6.02 4.84 4.84 5.01 0.42 6.02 4.28 6.02 4.55
# [9,] 4.80 4.84 5.01 4.80 6.35 4.55 4.80 6.02 4.80
# [,669] [,670] [,671] [,672] [,673] [,674] [,675] [,676] [,677]
# [1,] 5.01 4.84 5.21 4.80 4.55 5.21 4.28 0.42 0.42
# [2,] 4.28 0.42 4.80 4.80 4.28 4.28 4.80 5.21 5.01
# [3,] 4.28 4.84 5.21 4.84 5.21 4.80 0.42 6.02 4.84
# [4,] 6.02 4.84 4.84 4.84 0.42 4.28 4.28 4.28 6.35
# [5,] 5.01 6.02 0.42 4.28 4.28 5.21 4.55 4.28 6.02
# [6,] 6.35 4.28 4.84 4.80 4.55 4.80 5.01 4.80 6.35
# [7,] 6.02 4.84 4.84 6.02 5.01 4.84 6.02 5.21 6.35
# [8,] 6.35 6.02 4.80 4.55 5.01 4.80 6.35 4.84 4.28
# [9,] 4.28 4.80 0.42 0.42 4.28 0.42 4.28 6.02 4.80
# [,678] [,679] [,680] [,681] [,682] [,683] [,684] [,685] [,686]
# [1,] 4.80 0.42 4.80 4.84 4.84 6.02 6.35 4.55 5.21
# [2,] 4.55 4.55 6.02 5.21 6.35 4.28 6.02 4.55 4.55
# [3,] 6.02 4.55 0.42 4.28 6.02 4.28 4.55 4.28 5.01
# [4,] 4.80 0.42 6.02 4.84 4.28 5.01 6.02 4.28 4.55
# [5,] 4.80 6.02 6.35 4.28 4.28 4.28 5.01 5.21 0.42
# [6,] 6.02 6.02 5.01 4.28 4.55 6.35 4.55 5.01 5.21
# [7,] 6.35 4.80 6.35 4.80 0.42 6.35 6.35 0.42 6.02
# [8,] 4.80 4.84 6.35 4.28 6.02 4.80 0.42 4.55 0.42
# [9,] 4.84 5.01 4.84 6.35 6.35 6.02 4.28 4.55 5.01
# [,687] [,688] [,689] [,690] [,691] [,692] [,693] [,694] [,695]
# [1,] 6.02 4.80 6.02 4.80 4.80 0.42 4.55 0.42 4.28
# [2,] 4.55 4.28 0.42 6.35 5.01 6.35 6.02 4.55 6.35
# [3,] 4.55 6.35 6.35 5.01 4.80 4.55 4.84 6.02 6.02
# [4,] 4.80 6.35 6.35 5.21 5.21 4.84 6.02 6.35 6.02
# [5,] 4.28 5.21 4.55 6.35 6.02 0.42 0.42 5.01 4.28
# [6,] 4.80 4.84 4.55 6.35 5.01 6.02 5.21 4.84 4.55
# [7,] 5.01 4.84 4.80 5.01 5.01 4.28 6.35 6.35 6.35
# [8,] 5.01 0.42 4.55 5.21 4.55 6.02 6.35 5.21 5.21
# [9,] 5.01 4.28 6.35 5.01 4.80 4.55 4.80 4.55 6.02
# [,696] [,697] [,698] [,699] [,700] [,701] [,702] [,703] [,704]
# [1,] 4.84 6.02 6.35 4.80 4.80 5.21 5.21 4.80 4.80
# [2,] 5.01 4.55 4.84 4.55 4.84 5.21 4.80 5.21 4.55
# [3,] 6.35 4.55 4.80 4.28 5.01 0.42 4.84 4.84 4.55
# [4,] 6.02 4.28 5.01 0.42 4.55 5.21 4.28 0.42 6.02
# [5,] 5.01 5.01 6.35 4.84 5.01 4.28 4.55 0.42 5.21
# [6,] 4.80 5.01 4.55 6.02 5.21 4.84 4.28 4.55 6.35
# [7,] 0.42 5.21 4.84 6.35 5.21 4.80 5.21 4.84 4.80
# [8,] 5.21 4.28 6.35 6.35 4.80 5.21 0.42 4.28 5.21
# [9,] 4.80 6.35 6.35 4.80 4.28 6.02 4.28 5.21 4.84
# [,705] [,706] [,707] [,708] [,709] [,710] [,711] [,712] [,713]
# [1,] 6.02 5.21 4.28 0.42 4.55 4.28 0.42 5.01 5.21
# [2,] 4.84 4.55 5.01 5.21 5.01 5.01 4.80 6.35 6.02
# [3,] 5.01 0.42 5.21 6.35 4.55 4.84 4.28 0.42 5.01
# [4,] 6.35 4.28 4.55 6.35 4.55 6.35 4.55 6.02 4.80
# [5,] 6.35 5.01 5.21 4.80 4.84 6.35 0.42 6.35 4.84
# [6,] 4.55 5.21 4.80 4.28 4.28 6.02 0.42 4.80 0.42
# [7,] 0.42 0.42 4.28 4.28 5.01 4.55 5.21 5.21 4.55
# [8,] 5.21 4.80 0.42 4.84 4.84 4.55 4.55 6.02 4.80
# [9,] 6.02 5.01 4.84 4.55 5.01 0.42 6.35 0.42 6.35
# [,714] [,715] [,716] [,717] [,718] [,719] [,720] [,721] [,722]
# [1,] 4.28 6.35 6.35 6.02 0.42 4.80 4.28 4.84 4.84
# [2,] 4.55 5.01 4.80 0.42 5.01 0.42 4.55 5.21 5.01
# [3,] 5.21 6.02 0.42 0.42 5.01 0.42 0.42 5.21 4.80
# [4,] 4.80 6.02 4.55 6.35 6.02 6.02 4.55 4.80 4.55
# [5,] 0.42 5.21 5.21 4.80 4.80 5.01 5.21 6.02 4.28
# [6,] 4.84 4.84 4.28 5.21 5.21 4.80 6.02 5.21 5.01
# [7,] 6.35 4.28 4.55 6.02 0.42 4.80 4.80 4.28 5.01
# [8,] 6.35 4.84 5.21 5.01 4.84 6.35 6.02 5.01 6.02
# [9,] 6.35 5.21 4.80 6.35 4.28 4.84 4.84 4.84 0.42
# [,723] [,724] [,725] [,726] [,727] [,728] [,729] [,730] [,731]
# [1,] 4.84 5.21 4.80 6.02 5.21 4.84 4.80 4.28 4.55
# [2,] 4.80 0.42 4.28 4.84 4.55 6.02 4.80 6.35 4.55
# [3,] 5.01 5.21 4.80 6.35 5.21 5.01 4.28 4.55 4.28
# [4,] 4.80 4.84 4.28 6.35 5.21 6.35 0.42 5.21 5.01
# [5,] 5.01 5.21 4.55 6.35 5.21 0.42 6.02 4.84 6.02
# [6,] 6.02 6.35 4.55 4.84 4.80 4.28 5.21 6.02 5.21
# [7,] 4.84 6.35 4.28 5.01 6.02 6.02 6.35 4.28 4.55
# [8,] 4.28 0.42 5.21 4.80 4.28 5.21 4.28 4.80 4.80
# [9,] 4.84 4.55 4.28 5.21 5.01 4.28 6.02 4.84 6.02
# [,732] [,733] [,734] [,735] [,736] [,737] [,738] [,739] [,740]
# [1,] 6.35 4.55 6.35 4.28 5.21 4.55 4.84 5.01 4.84
# [2,] 4.55 4.55 4.84 4.80 4.28 4.80 6.35 5.21 4.55
# [3,] 5.21 4.84 4.80 5.21 5.21 6.35 4.28 4.55 5.21
# [4,] 6.02 4.55 4.84 5.01 6.35 6.35 0.42 4.28 4.55
# [5,] 4.80 4.80 5.21 4.84 6.35 4.55 6.02 6.35 5.21
# [6,] 6.02 4.55 5.01 4.55 4.55 4.80 4.84 5.01 6.02
# [7,] 5.21 5.21 5.21 6.02 6.35 4.28 4.28 0.42 5.01
# [8,] 5.21 4.80 4.80 0.42 0.42 4.55 5.01 6.35 4.80
# [9,] 5.21 5.01 5.21 4.80 6.02 5.21 5.01 5.21 4.80
# [,741] [,742] [,743] [,744] [,745] [,746] [,747] [,748] [,749]
# [1,] 4.80 4.55 5.01 5.01 5.01 6.35 4.84 4.28 0.42
# [2,] 5.01 4.84 6.02 6.35 4.80 5.21 0.42 4.28 4.80
# [3,] 0.42 5.01 4.80 4.80 0.42 4.55 6.02 4.84 4.80
# [4,] 6.35 5.21 5.21 6.35 6.35 0.42 4.84 6.02 5.21
# [5,] 6.35 5.01 5.01 5.01 4.28 0.42 0.42 4.55 0.42
# [6,] 0.42 4.84 5.01 6.35 5.01 0.42 4.55 5.01 4.28
# [7,] 4.84 6.02 0.42 5.21 0.42 0.42 5.01 4.80 4.80
# [8,] 4.80 6.35 4.28 5.01 4.80 5.01 4.55 4.28 5.21
# [9,] 6.35 5.21 4.55 6.02 4.55 6.02 4.84 4.55 5.01
# [,750] [,751] [,752] [,753] [,754] [,755] [,756] [,757] [,758]
# [1,] 4.28 6.35 5.21 4.55 4.28 6.02 5.01 4.80 4.84
# [2,] 5.01 5.21 4.84 5.21 6.02 4.28 5.21 4.80 6.02
# [3,] 4.80 4.80 5.01 6.02 4.84 6.35 5.21 4.84 6.35
# [4,] 6.02 4.55 4.55 6.35 5.01 0.42 4.55 5.21 6.02
# [5,] 4.28 4.28 4.84 6.02 6.02 5.21 6.35 4.28 5.21
# [6,] 6.35 6.35 6.35 6.35 4.28 0.42 4.80 0.42 0.42
# [7,] 0.42 4.55 4.80 6.02 5.01 6.02 0.42 5.21 4.84
# [8,] 5.21 4.84 4.84 6.02 4.55 4.80 5.01 4.80 4.28
# [9,] 4.28 5.01 4.55 5.01 6.35 5.01 6.02 6.35 5.01
# [,759] [,760] [,761] [,762] [,763] [,764] [,765] [,766] [,767]
# [1,] 6.35 4.80 4.84 4.55 6.02 4.28 6.02 5.01 6.02
# [2,] 4.55 6.02 6.35 5.01 4.55 6.02 4.80 6.02 4.55
# [3,] 4.55 6.35 4.55 5.01 4.55 4.84 4.55 4.28 4.55
# [4,] 5.21 6.02 4.84 6.02 4.84 4.55 0.42 0.42 4.28
# [5,] 0.42 4.28 5.21 4.55 5.21 4.80 4.55 5.21 4.55
# [6,] 6.35 5.01 6.35 4.55 6.02 4.28 4.84 4.55 4.80
# [7,] 6.35 5.01 4.80 4.28 4.84 6.02 4.84 4.55 5.01
# [8,] 4.80 5.01 5.21 4.84 4.55 4.84 5.21 6.35 4.80
# [9,] 5.21 4.80 5.21 4.80 4.80 6.02 6.35 5.01 6.02
# [,768] [,769] [,770] [,771] [,772] [,773] [,774] [,775] [,776]
# [1,] 5.01 6.02 4.84 4.55 4.84 0.42 6.02 0.42 4.84
# [2,] 5.21 4.55 4.55 5.01 6.35 6.02 6.35 5.01 4.55
# [3,] 4.28 5.21 6.02 5.21 6.02 4.80 6.02 5.01 6.35
# [4,] 0.42 6.35 0.42 6.02 4.28 5.21 6.35 6.35 4.28
# [5,] 5.21 4.84 4.84 4.28 5.21 4.84 0.42 4.84 4.55
# [6,] 5.01 4.84 6.02 4.55 6.02 0.42 0.42 5.21 4.28
# [7,] 0.42 5.01 4.55 4.55 5.21 5.21 4.28 4.28 5.01
# [8,] 6.35 4.84 6.35 4.55 4.28 4.84 5.01 0.42 4.28
# [9,] 6.02 6.02 6.35 4.55 0.42 4.80 0.42 5.21 4.28
# [,777] [,778] [,779] [,780] [,781] [,782] [,783] [,784] [,785]
# [1,] 6.02 4.80 6.02 6.35 4.84 5.21 6.35 0.42 0.42
# [2,] 4.28 6.35 5.01 6.35 6.02 4.28 6.35 4.80 6.02
# [3,] 4.80 4.84 4.28 5.01 5.01 4.80 0.42 0.42 4.55
# [4,] 4.55 4.80 4.28 6.02 4.80 4.80 4.80 6.02 4.28
# [5,] 6.35 5.21 5.01 4.55 4.84 6.35 4.84 5.21 4.80
# [6,] 5.01 6.02 5.21 4.80 0.42 6.35 5.01 5.01 6.35
# [7,] 0.42 4.28 0.42 5.01 4.55 4.80 4.80 4.55 6.02
# [8,] 4.80 4.80 4.55 5.21 5.01 5.01 4.80 0.42 4.84
# [9,] 6.35 5.01 6.35 4.84 5.01 6.35 5.21 4.55 4.28
# [,786] [,787] [,788] [,789] [,790] [,791] [,792] [,793] [,794]
# [1,] 4.55 6.02 6.35 5.21 4.80 6.35 0.42 4.80 4.28
# [2,] 0.42 4.84 6.02 5.01 4.28 4.28 4.84 0.42 6.35
# [3,] 4.55 0.42 6.02 0.42 6.02 6.35 4.80 4.84 4.28
# [4,] 6.35 6.02 4.84 4.84 0.42 6.02 4.28 6.02 6.35
# [5,] 4.80 6.35 4.80 4.80 6.35 4.84 6.02 0.42 6.02
# [6,] 4.28 5.01 5.21 6.35 6.35 0.42 4.28 5.01 4.28
# [7,] 6.02 6.02 6.02 5.21 0.42 6.02 0.42 6.02 5.21
# [8,] 0.42 4.55 4.84 4.80 4.80 5.21 5.01 4.84 5.01
# [9,] 4.80 4.28 6.35 6.02 5.21 4.80 4.84 5.01 5.21
# [,795] [,796] [,797] [,798] [,799] [,800] [,801] [,802] [,803]
# [1,] 4.80 4.80 5.01 5.21 0.42 6.35 5.21 5.21 5.01
# [2,] 4.28 4.84 4.84 0.42 4.28 6.02 5.01 6.02 4.55
# [3,] 4.28 4.28 6.35 6.02 5.01 0.42 4.84 4.80 6.35
# [4,] 4.28 0.42 4.55 6.35 4.80 4.55 4.84 4.28 5.01
# [5,] 4.80 5.21 6.02 4.55 5.21 4.84 4.28 5.21 6.02
# [6,] 6.02 4.55 4.28 5.01 6.02 5.21 5.21 4.28 4.55
# [7,] 6.35 4.84 4.84 0.42 0.42 4.80 6.02 0.42 4.55
# [8,] 6.02 6.02 4.55 5.01 5.21 6.02 4.28 5.01 0.42
# [9,] 5.01 4.84 6.02 6.35 5.01 6.02 4.80 5.01 5.21
# [,804] [,805] [,806] [,807] [,808] [,809] [,810] [,811] [,812]
# [1,] 5.21 4.84 6.02 4.28 0.42 4.55 6.35 4.28 0.42
# [2,] 5.21 6.02 4.55 6.35 6.02 6.35 5.21 4.55 4.80
# [3,] 5.01 0.42 0.42 6.35 5.21 4.28 4.28 0.42 5.21
# [4,] 0.42 5.01 4.55 4.55 6.35 4.55 5.01 4.80 4.80
# [5,] 6.02 6.02 6.35 5.01 6.35 5.01 4.55 0.42 0.42
# [6,] 5.01 5.21 5.01 0.42 5.21 4.28 4.80 4.55 6.02
# [7,] 4.84 6.02 4.28 4.55 0.42 5.21 5.01 0.42 4.28
# [8,] 6.35 0.42 4.55 4.55 5.01 4.84 0.42 4.28 4.55
# [9,] 4.55 0.42 5.21 0.42 5.01 6.35 0.42 5.01 4.84
# [,813] [,814] [,815] [,816] [,817] [,818] [,819] [,820] [,821]
# [1,] 5.01 0.42 5.01 4.28 6.35 4.80 4.80 6.02 4.84
# [2,] 6.02 6.02 6.02 4.84 6.02 0.42 5.21 4.28 5.01
# [3,] 4.55 0.42 4.55 4.80 5.21 5.01 6.02 0.42 4.55
# [4,] 4.80 4.84 5.01 5.01 4.55 5.01 4.28 6.35 4.84
# [5,] 4.28 6.02 5.01 4.28 4.28 5.01 5.21 4.28 4.84
# [6,] 5.21 6.02 4.55 0.42 4.80 6.35 5.21 4.80 4.80
# [7,] 5.01 5.21 4.28 4.84 4.28 5.21 4.84 4.55 6.35
# [8,] 5.21 4.80 4.28 4.55 5.01 5.21 4.84 4.80 4.80
# [9,] 4.80 4.55 4.80 4.28 0.42 6.35 4.55 0.42 5.01
# [,822] [,823] [,824] [,825] [,826] [,827] [,828] [,829] [,830]
# [1,] 6.35 4.55 4.84 4.80 6.35 4.28 6.35 5.21 5.21
# [2,] 5.01 6.35 4.28 4.80 6.02 5.01 4.28 4.80 4.28
# [3,] 6.35 4.28 4.28 4.55 4.28 5.01 4.84 5.01 4.80
# [4,] 4.55 4.80 4.55 4.55 5.01 4.28 6.35 4.28 4.55
# [5,] 5.21 5.01 5.21 6.02 4.80 4.84 6.35 4.55 0.42
# [6,] 6.35 6.35 4.55 4.80 4.28 4.80 4.80 5.21 5.01
# [7,] 4.55 6.35 4.80 4.84 4.80 0.42 6.02 4.28 4.55
# [8,] 5.21 6.35 4.28 4.84 4.80 4.55 4.55 4.84 4.80
# [9,] 4.84 6.35 0.42 5.01 0.42 4.55 5.21 6.35 0.42
# [,831] [,832] [,833] [,834] [,835] [,836] [,837] [,838] [,839]
# [1,] 6.35 4.84 4.84 4.80 4.80 4.55 6.35 5.01 4.55
# [2,] 6.35 6.02 6.35 6.35 4.80 6.02 6.02 4.55 6.02
# [3,] 4.55 5.01 6.35 6.35 4.28 4.28 5.01 5.21 5.21
# [4,] 5.01 6.35 4.55 0.42 0.42 6.35 0.42 4.55 4.80
# [5,] 4.84 6.35 5.21 0.42 6.02 0.42 4.80 5.01 4.84
# [6,] 6.02 4.55 5.01 4.80 4.28 4.55 6.35 6.02 4.80
# [7,] 4.55 0.42 5.01 4.28 5.01 4.84 4.28 4.80 6.02
# [8,] 4.55 4.55 4.28 4.84 4.84 4.84 4.80 0.42 0.42
# [9,] 4.28 4.28 5.21 4.80 5.01 4.80 4.80 6.02 4.55
# [,840] [,841] [,842] [,843] [,844] [,845] [,846] [,847] [,848]
# [1,] 0.42 0.42 4.80 4.80 4.80 4.84 4.28 4.80 0.42
# [2,] 0.42 4.80 4.84 5.21 4.80 4.55 5.21 5.01 4.84
# [3,] 4.80 4.55 4.80 4.55 5.21 4.28 0.42 5.21 4.80
# [4,] 4.55 6.35 6.35 6.02 0.42 4.84 5.21 4.84 5.21
# [5,] 6.02 6.35 5.01 4.84 5.01 6.02 4.84 4.28 6.35
# [6,] 0.42 5.21 5.21 6.35 5.21 0.42 5.21 4.84 4.55
# [7,] 5.01 4.84 6.02 4.28 4.80 6.35 4.80 4.28 5.21
# [8,] 4.28 0.42 4.28 0.42 5.21 4.28 5.21 0.42 4.84
# [9,] 5.01 4.84 6.35 6.02 6.35 5.01 5.01 5.01 0.42
# [,849] [,850] [,851] [,852] [,853] [,854] [,855] [,856] [,857]
# [1,] 0.42 4.80 5.01 5.01 6.35 6.35 6.35 6.02 4.28
# [2,] 4.28 5.01 0.42 6.35 6.02 4.84 4.28 5.21 5.01
# [3,] 4.55 6.35 4.80 6.35 5.21 5.01 4.80 5.01 0.42
# [4,] 6.02 4.28 6.02 0.42 6.02 4.55 6.35 0.42 5.01
# [5,] 6.35 4.55 4.28 0.42 4.55 4.80 4.84 5.01 5.01
# [6,] 4.55 4.55 6.02 6.02 0.42 5.21 4.80 0.42 6.02
# [7,] 6.02 6.35 4.55 4.28 0.42 0.42 0.42 4.28 6.02
# [8,] 5.21 0.42 4.28 6.02 6.02 4.28 4.80 4.80 4.84
# [9,] 0.42 0.42 6.35 4.28 4.84 4.28 6.35 4.55 5.21
# [,858] [,859] [,860] [,861] [,862] [,863] [,864] [,865] [,866]
# [1,] 4.84 4.80 6.35 0.42 6.35 6.02 4.28 0.42 4.28
# [2,] 4.28 6.35 5.21 4.84 0.42 4.28 4.80 6.35 4.28
# [3,] 4.55 4.28 6.35 6.02 0.42 0.42 4.55 6.02 6.02
# [4,] 4.55 4.80 4.84 0.42 4.28 4.80 6.35 4.28 6.02
# [5,] 4.55 6.02 6.35 6.02 6.35 4.28 6.35 4.84 6.35
# [6,] 4.55 6.35 0.42 5.01 0.42 4.80 4.84 0.42 0.42
# [7,] 4.28 4.80 5.21 4.84 4.84 4.28 4.80 4.28 5.21
# [8,] 4.55 0.42 0.42 4.55 0.42 4.80 6.02 4.55 6.02
# [9,] 4.80 5.21 4.80 0.42 5.21 4.84 0.42 4.84 6.35
# [,867] [,868] [,869] [,870] [,871] [,872] [,873] [,874] [,875]
# [1,] 5.01 4.80 5.01 4.84 4.80 6.02 5.01 4.80 4.84
# [2,] 4.80 4.55 5.01 4.55 4.55 4.55 5.21 5.01 4.80
# [3,] 4.84 5.01 6.02 0.42 4.84 0.42 5.21 6.35 0.42
# [4,] 5.01 4.28 5.21 5.21 4.84 6.35 5.01 6.02 6.35
# [5,] 6.02 6.35 5.21 4.28 4.28 5.21 4.84 4.28 0.42
# [6,] 0.42 4.28 4.84 4.84 4.80 6.02 5.21 0.42 4.84
# [7,] 5.01 0.42 0.42 4.80 6.35 5.21 6.35 5.21 4.80
# [8,] 4.84 5.01 6.02 6.35 5.21 0.42 4.80 4.28 5.01
# [9,] 0.42 4.28 5.01 4.55 4.28 5.01 0.42 0.42 4.28
# [,876] [,877] [,878] [,879] [,880] [,881] [,882] [,883] [,884]
# [1,] 6.35 4.80 5.21 5.21 4.84 4.84 0.42 6.35 4.55
# [2,] 0.42 0.42 4.55 6.02 4.55 4.84 5.01 5.01 0.42
# [3,] 5.01 6.02 4.55 4.28 4.55 4.28 5.01 5.21 5.01
# [4,] 5.01 4.80 5.01 6.02 6.02 5.21 4.28 4.55 4.55
# [5,] 0.42 4.55 4.55 4.55 4.84 4.80 4.80 6.02 4.28
# [6,] 6.02 6.35 4.84 4.28 4.80 4.55 6.02 0.42 5.21
# [7,] 4.28 4.28 6.35 5.21 5.21 6.35 5.21 4.55 5.21
# [8,] 5.01 4.28 6.35 4.28 4.55 4.84 4.84 4.84 5.01
# [9,] 6.02 6.35 5.01 4.28 4.55 6.02 4.55 4.80 4.84
# [,885] [,886] [,887] [,888] [,889] [,890] [,891] [,892] [,893]
# [1,] 4.84 4.84 5.21 5.21 4.28 0.42 4.84 4.80 4.28
# [2,] 4.55 4.28 4.55 4.28 4.80 6.02 4.55 6.35 4.80
# [3,] 6.02 5.21 4.84 4.80 4.55 0.42 4.84 6.35 0.42
# [4,] 5.21 5.01 5.01 4.55 4.84 4.28 0.42 5.21 5.01
# [5,] 5.21 5.01 4.55 4.55 4.28 4.80 4.55 4.80 6.02
# [6,] 5.01 4.55 4.80 4.84 4.55 4.84 6.02 0.42 4.84
# [7,] 0.42 4.84 4.80 4.55 6.35 4.84 4.55 4.55 6.02
# [8,] 4.55 6.35 6.02 5.01 4.84 4.55 5.21 4.28 0.42
# [9,] 5.21 5.01 0.42 6.02 5.21 5.21 4.28 4.55 4.55
# [,894] [,895] [,896] [,897] [,898] [,899] [,900] [,901] [,902]
# [1,] 5.21 5.21 6.02 4.28 6.02 4.55 4.84 4.80 0.42
# [2,] 5.01 4.28 5.01 6.35 4.55 6.02 5.21 6.35 6.35
# [3,] 6.35 5.21 5.01 5.01 5.21 4.28 4.28 5.21 4.84
# [4,] 0.42 5.21 0.42 6.02 4.55 6.02 5.01 5.01 4.28
# [5,] 5.21 6.35 5.21 4.80 5.21 6.02 6.35 4.28 6.35
# [6,] 6.02 4.84 4.28 6.02 4.28 4.84 0.42 5.21 0.42
# [7,] 4.28 0.42 0.42 4.55 4.28 5.21 4.84 6.35 5.21
# [8,] 4.28 5.01 4.84 6.02 4.55 4.28 5.01 4.55 0.42
# [9,] 6.35 0.42 4.28 4.84 4.55 6.35 4.84 6.02 0.42
# [,903] [,904] [,905] [,906] [,907] [,908] [,909] [,910] [,911]
# [1,] 5.21 0.42 0.42 4.80 5.01 5.21 4.84 6.02 4.28
# [2,] 0.42 4.28 4.55 4.28 4.28 6.35 4.84 4.28 4.55
# [3,] 4.80 4.80 4.55 5.01 5.01 6.35 6.35 4.55 4.55
# [4,] 4.28 4.80 5.01 6.02 5.21 6.02 4.55 4.55 0.42
# [5,] 6.35 4.28 5.21 4.28 5.21 5.01 4.55 4.80 0.42
# [6,] 4.80 4.55 0.42 6.02 6.02 6.02 4.28 5.21 6.02
# [7,] 4.55 5.01 6.02 4.80 5.21 4.80 4.55 4.55 4.80
# [8,] 5.01 6.35 4.28 0.42 6.35 5.01 4.55 4.80 4.80
# [9,] 4.84 6.35 6.02 4.80 5.01 4.80 0.42 6.35 4.84
# [,912] [,913] [,914] [,915] [,916] [,917] [,918] [,919] [,920]
# [1,] 6.02 5.01 6.35 4.80 4.80 4.28 5.21 6.35 4.55
# [2,] 4.84 4.55 4.80 5.01 6.02 0.42 4.28 6.02 4.80
# [3,] 4.84 5.01 4.28 5.01 6.35 5.21 6.35 4.55 4.80
# [4,] 4.80 0.42 4.84 4.55 6.35 4.28 4.55 0.42 4.84
# [5,] 6.35 4.84 6.02 6.02 4.55 4.55 4.80 4.80 6.35
# [6,] 6.02 4.55 0.42 4.28 6.02 5.01 5.01 6.02 4.28
# [7,] 6.35 0.42 6.35 4.28 5.01 6.02 5.21 6.35 4.55
# [8,] 4.80 5.01 4.55 4.55 4.84 5.01 6.02 4.80 0.42
# [9,] 6.35 5.01 4.80 6.02 5.01 5.01 5.21 6.35 4.80
# [,921] [,922] [,923] [,924] [,925] [,926] [,927] [,928] [,929]
# [1,] 4.28 5.21 5.21 4.28 5.21 6.02 0.42 5.01 5.01
# [2,] 4.84 0.42 5.01 6.02 6.02 4.28 6.02 6.35 6.35
# [3,] 4.80 4.80 4.80 5.21 5.01 4.80 5.01 6.35 4.84
# [4,] 6.35 6.35 0.42 4.55 6.35 4.84 4.28 4.80 4.55
# [5,] 6.02 4.55 5.01 4.28 5.01 5.01 4.55 4.80 4.55
# [6,] 0.42 5.21 5.21 6.02 4.80 4.84 4.80 4.84 5.21
# [7,] 4.80 6.35 4.55 4.84 5.01 6.35 4.55 4.28 6.35
# [8,] 4.55 6.35 4.55 5.21 5.01 4.55 5.01 0.42 6.35
# [9,] 0.42 4.28 4.55 4.84 4.55 0.42 5.21 4.28 4.55
# [,930] [,931] [,932] [,933] [,934] [,935] [,936] [,937] [,938]
# [1,] 4.55 4.84 5.21 0.42 5.01 4.55 4.80 4.84 5.21
# [2,] 4.84 6.02 6.35 6.35 5.01 4.84 4.84 5.01 6.02
# [3,] 5.01 6.02 4.80 4.84 0.42 5.21 4.28 4.28 4.80
# [4,] 0.42 5.21 4.80 4.28 4.80 6.35 6.35 5.01 6.35
# [5,] 6.02 6.35 6.35 6.35 4.80 4.80 4.28 6.35 4.55
# [6,] 0.42 4.28 4.84 0.42 4.28 0.42 4.28 4.84 0.42
# [7,] 4.84 4.55 4.55 4.55 6.35 6.02 4.84 4.28 4.28
# [8,] 5.21 0.42 4.84 4.84 4.84 6.02 4.55 4.55 6.02
# [9,] 6.35 6.35 0.42 4.80 4.84 0.42 4.80 5.21 4.84
# [,939] [,940] [,941] [,942] [,943] [,944] [,945] [,946] [,947]
# [1,] 6.35 6.02 5.21 6.35 0.42 5.01 4.55 4.55 4.80
# [2,] 4.55 4.55 5.01 6.02 5.21 5.01 4.28 4.28 0.42
# [3,] 4.80 4.28 6.35 4.84 4.55 5.21 0.42 0.42 0.42
# [4,] 4.55 5.01 5.01 4.55 4.84 5.21 4.55 0.42 4.80
# [5,] 4.84 5.21 4.80 6.02 5.21 4.28 4.80 4.55 0.42
# [6,] 6.02 6.02 0.42 6.02 6.35 4.80 4.80 0.42 6.02
# [7,] 6.02 4.55 4.28 5.21 5.01 6.02 6.02 5.21 4.84
# [8,] 4.80 5.21 4.55 5.01 4.84 0.42 4.55 5.21 4.28
# [9,] 4.80 4.80 4.28 4.84 6.35 5.01 6.35 4.80 0.42
# [,948] [,949] [,950] [,951] [,952] [,953] [,954] [,955] [,956]
# [1,] 5.01 0.42 6.02 5.21 4.55 6.35 4.55 4.84 4.84
# [2,] 5.01 5.21 4.55 5.21 6.35 6.02 4.28 5.01 4.28
# [3,] 5.01 4.80 6.35 5.01 4.84 0.42 6.35 4.28 4.28
# [4,] 4.84 4.55 4.28 4.55 0.42 4.28 5.01 5.01 0.42
# [5,] 6.35 5.01 4.80 4.55 4.55 5.21 6.02 4.80 5.01
# [6,] 5.01 4.28 4.80 5.01 4.80 4.55 4.55 4.55 4.84
# [7,] 4.55 4.84 4.84 4.80 5.21 4.55 4.84 0.42 6.35
# [8,] 0.42 0.42 4.80 4.55 4.80 0.42 4.84 6.02 4.80
# [9,] 6.02 4.55 5.01 0.42 4.84 4.55 6.35 6.35 4.84
# [,957] [,958] [,959] [,960] [,961] [,962] [,963] [,964] [,965]
# [1,] 4.84 4.28 5.01 4.28 4.55 4.28 6.35 5.01 0.42
# [2,] 5.21 4.84 6.35 0.42 4.84 0.42 0.42 4.84 5.01
# [3,] 6.35 4.28 4.84 4.84 4.55 0.42 5.01 6.35 5.21
# [4,] 4.80 6.02 0.42 4.84 4.28 5.01 4.80 4.55 5.21
# [5,] 6.02 6.35 5.21 6.02 4.28 6.35 0.42 4.55 0.42
# [6,] 6.02 6.35 0.42 4.80 5.01 4.80 5.01 4.55 4.55
# [7,] 4.55 4.55 6.02 6.35 4.84 4.55 6.02 4.84 6.35
# [8,] 4.55 5.01 4.80 5.01 4.80 4.28 4.55 4.84 5.01
# [9,] 6.35 4.28 4.80 4.28 5.01 6.02 4.80 6.02 4.84
# [,966] [,967] [,968] [,969] [,970] [,971] [,972] [,973] [,974]
# [1,] 4.28 6.35 6.02 6.35 0.42 4.84 4.80 5.01 5.21
# [2,] 4.55 5.01 6.02 0.42 5.21 5.21 4.80 4.28 5.01
# [3,] 6.35 4.28 4.55 4.28 4.84 6.35 5.21 6.35 5.21
# [4,] 6.35 5.21 4.28 4.28 5.01 4.84 5.01 4.84 4.84
# [5,] 0.42 0.42 4.80 6.35 4.80 5.01 4.84 5.01 5.01
# [6,] 4.84 5.21 5.21 4.28 5.21 4.28 6.02 5.01 0.42
# [7,] 5.01 5.01 5.01 6.02 6.35 5.21 5.21 0.42 5.01
# [8,] 5.21 0.42 4.55 0.42 4.84 0.42 4.28 4.55 0.42
# [9,] 6.02 6.35 6.35 5.21 5.21 0.42 0.42 4.80 6.35
# [,975] [,976] [,977] [,978] [,979] [,980] [,981] [,982] [,983]
# [1,] 6.02 5.21 4.80 6.02 6.02 5.01 5.21 6.02 4.28
# [2,] 0.42 4.84 6.02 0.42 4.80 4.28 4.28 6.35 4.55
# [3,] 4.80 4.80 6.02 6.02 4.80 4.80 4.80 5.21 4.80
# [4,] 0.42 5.01 6.35 6.02 0.42 6.02 4.80 6.35 6.02
# [5,] 4.28 4.28 0.42 4.80 5.01 4.28 4.80 6.02 6.35
# [6,] 4.55 4.84 5.01 5.01 0.42 6.35 4.28 4.80 4.80
# [7,] 5.01 4.55 4.28 4.84 4.55 4.55 6.02 5.01 4.28
# [8,] 4.28 6.02 4.55 6.02 4.84 4.80 4.28 4.80 0.42
# [9,] 5.21 5.21 6.02 6.35 6.35 5.01 6.35 4.55 4.28
# [,984] [,985] [,986] [,987] [,988] [,989] [,990] [,991] [,992]
# [1,] 4.55 0.42 6.35 0.42 0.42 6.02 4.80 6.35 6.35
# [2,] 4.84 6.02 4.55 5.01 5.21 4.55 4.84 4.28 5.21
# [3,] 4.84 5.01 6.35 4.55 4.84 4.55 5.21 6.02 5.21
# [4,] 5.21 6.02 5.01 6.35 5.21 4.80 4.55 4.84 4.80
# [5,] 6.02 5.01 6.35 6.02 5.01 4.28 4.80 6.35 4.55
# [6,] 4.80 0.42 6.02 4.80 6.35 4.80 5.01 6.02 4.84
# [7,] 5.21 6.02 4.55 5.21 4.55 6.35 4.28 6.35 4.80
# [8,] 4.55 5.21 4.84 0.42 5.21 5.21 4.28 5.21 4.28
# [9,] 6.35 0.42 6.02 4.84 6.02 6.02 6.35 6.02 4.80
# [,993] [,994] [,995] [,996] [,997] [,998] [,999] [,1000]
# [1,] 4.28 4.55 6.35 4.28 4.55 4.28 4.80 0.42
# [2,] 4.28 6.02 6.02 5.01 4.55 6.02 5.01 4.80
# [3,] 6.35 4.80 4.84 4.84 0.42 6.35 4.84 6.35
# [4,] 6.35 4.80 4.55 4.84 0.42 4.28 4.80 4.55
# [5,] 6.35 6.35 0.42 6.35 0.42 6.35 6.02 0.42
# [6,] 5.21 4.55 6.02 6.02 6.02 5.01 5.21 4.55
# [7,] 5.01 4.28 4.84 4.28 5.01 0.42 5.21 4.80
# [8,] 5.21 4.28 5.21 4.55 4.80 6.35 6.35 5.01
# [9,] 6.35 4.28 6.35 4.28 4.28 0.42 5.01 4.55
DEPARTMENT OF MATHEMATICS & STATISTICS 44
Compute the bootstrap estimates
mu1 = apply(Boot, 2, function(x) sort(x)[5])
mu2 = apply(Boot, 2, function(x) mean(x))
or can use dplyr package to manipulate the data.
library(dplyr)
# Make data.frame
dat = data.frame(b = as.vector(Boot),
Sample = rep(1:m, rep(n, m)))
# Compute resampled estimates
bootEsts = dat %>%
group_by(Sample) %>%
summarise(mu_1 = sort(b)[5], mu_2 = mean(b))
DEPARTMENT OF MATHEMATICS & STATISTICS 45
View estimates
bootEsts
# # A tibble: 1,000 x 3
# Sample mu_1 mu_2
#
# 1 1 4.8 4.49
# 2 2 5.21 5.32
# 3 3 4.28 3.25
# 4 4 5.21 4.61
# 5 5 4.28 3.41
# 6 6 4.84 4.44
# 7 7 5.01 5.32
# 8 8 4.8 4.72
# 9 9 5.21 5.35
# 10 10 5.01 4.88
# # ... with 990 more rows
DEPARTMENT OF MATHEMATICS & STATISTICS 46
Compute statistical measures from samples
library(tidyr)
nonEsts = bootEsts %>%
gather(type, estimate, mu_1:mu_2) %>%
mutate(thetaHat = if_else(type == "mu_1", med, mn)) %>%
group_by(type) %>%
summarise(estMean = mean(estimate),
bias = mean(estimate - thetaHat),
var = var(estimate),
mse = mean((estimate - thetaHat)^2))
nonEsts
# # A tibble: 2 x 5
# type estMean bias var mse
#
# 1 mu_1 4.91 0.0690 0.0830 0.0877
# 2 mu_2 4.62 0.0130 0.279 0.279
I It appears µ̂1 = X˜ is superior to µ̂2 = X to estimate µ.
DEPARTMENT OF MATHEMATICS & STATISTICS 47
Parametric bootstrap
I The parametric bootstrap is the same procedure as the nonparametric
bootstrap but with a crucial change.
I A parametric distribution is used to resample observations instead of
the nonparametric ECDF.
I Amended procedure, assuming Xi ∼ fX (·, θ)
1. Compute the estimate θ̂ = t(x)
2. Draw a sample from fX (·, θ̂)
I Denote this new (bootstrap) sample x i∗ =
(
x1∗ x2∗ . . . xn∗
)T
I Compute a new estimate of θ denoted θ̂1∗ = t(x i∗), using the
bootstrap sample above.
3. Repeat step 2. m times to obtain a set of bootstrap estimates
I (θ̂1∗ θ̂2∗ . . . θ̂m∗)
4. Inspect the distribution of the bootstrap estimates to determine
properties of θ̂.
I Or compute summary properties from the induced bootstrapped
distribution.
DEPARTMENT OF MATHEMATICS & STATISTICS 48































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































# 1 1 4.8 4.49
# 2 2 5.21 5.32
# 3 3 4.28 3.25
# 4 4 5.21 4.61
# 5 5 4.28 3.41
# 6 6 4.84 4.44
# 7 7 5.01 5.32
# 8 8 4.8 4.72
# 9 9 5.21 5.35
# 10 10 5.01 4.88
# # ... with 990 more rows
DEPARTMENT OF MATHEMATICS & STATISTICS 46
Compute statistical measures from samples
library(tidyr)
nonEsts = bootEsts %>%
gather(type, estimate, mu_1:mu_2) %>%
mutate(thetaHat = if_else(type == "mu_1", med, mn)) %>%
group_by(type) %>%
summarise(estMean = mean(estimate),
bias = mean(estimate - thetaHat),
var = var(estimate),
mse = mean((estimate - thetaHat)^2))
nonEsts
# # A tibble: 2 x 5
# type estMean bias var mse
#
# 1 mu_1 4.91 0.0690 0.0830 0.0877
# 2 mu_2 4.62 0.0130 0.279 0.279
I It appears µ̂1 = X˜ is superior to µ̂2 = X to estimate µ.
DEPARTMENT OF MATHEMATICS & STATISTICS 47
Parametric bootstrap
I The parametric bootstrap is the same procedure as the nonparametric
bootstrap but with a crucial change.
I A parametric distribution is used to resample observations instead of
the nonparametric ECDF.
I Amended procedure, assuming Xi ∼ fX (·, θ)
1. Compute the estimate θ̂ = t(x)
2. Draw a sample from fX (·, θ̂)
I Denote this new (bootstrap) sample x i∗ =
(
x1∗ x2∗ . . . xn∗
)T
I Compute a new estimate of θ denoted θ̂1∗ = t(x i∗), using the
bootstrap sample above.
3. Repeat step 2. m times to obtain a set of bootstrap estimates
I (θ̂1∗ θ̂2∗ . . . θ̂m∗)
4. Inspect the distribution of the bootstrap estimates to determine
properties of θ̂.
I Or compute summary properties from the induced bootstrapped
distribution.
DEPARTMENT OF MATHEMATICS & STATISTICS 48

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