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统计代写-JUNE 2006

时间：2021-05-02

THE UNIVERSITY OF NEW SOUTH WALES

SCHOOL OF MATHEMATICS

FINAL EXAMINATION

JUNE 2006

MATH3811

STATISTICAL INFERENCE

(1) TIME ALLOWED – 2 Hours

(2) TOTAL NUMBER OF QUESTIONS – 4

(3) ANSWER ALL QUESTIONS

(4) THE QUESTIONS ARE NOT OF EQUAL VALUE

(5) TOTAL NUMBER OF MARKS – 100

(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE

All answers must be written in ink. Except where they are expressly required pencils

may only be used for drawing, sketching or graphical work.

JUNE 2006 MATH3811 Page 2

1. [28 marks] Suppose a sample X = (X1, X2, . . . , Xn) from the density

f(x, θ) =

{

(1 + θ)xθ, x ∈ (0, 1),

0 elsewhere

is given where θ > −1 is the unknown parameter.

a) Calculate E(X1).

b) Find a minimal sufficient statistic T for θ. Give reasons for your answer.

c) Calculate the Fisher information about θ contained in the statistic T that

you found in b). Show that it is equal to n

(1+θ)2

.

d) Find the MLE of θ and also the MLE of h(θ) = 1

1+θ

.

e) Find the Cramer-Rao lower bound for the variance of an unbiased esti-

mator of h(θ) = 1

1+θ

.

f) Write down the score function. Find out if an UMVUE of h(θ) = 1

1+θ

exists and if it exists, write it down. Is there an unbiased estimator of

h(θ) = θ whose variance attains the Cramer-Rao bound? Explain your

answers.

g) State the asymptotic distribution of

√

n(hˆmle − h(θ)) for h(θ) = 11+θ .

Hint: The “delta method” implies that for any smooth function h(θ) :

√

n(h(θˆmle)− h(θ0))→d N(0, (∂h

∂θ

(θ0))

2I−1(θ0)),

I(θ) = E{ ∂

∂θ

[lnf(x, θ)]}2 = E{− ∂

2

∂θ2

[lnf(x, θ)]}.

2. [28 marks]

a) Let X = (X1, X2, . . . , Xn) be i.i.d. observations, each from a Poisson

distribution:

f(y|λ) = e(−λ) · λy/(y!), y = 0, 1, 2, . . . , λ > 0.

The prior on λ is believed to be Gamma(2,3).

(Note that, for any values of α > 0, β > 0, the density of the Gamma(α, β)

distribution is given by

τ(λ) =

{

1

βα·Γ(α) · λα−1 exp(−λ/β) , λ > 0;

0 else

and Γ(α) =

∫∞

0 e

−xxα−1dx is the Gamma function with the property

Γ(α+ 1) = αΓ(α).)

Please see over . . .

JUNE 2006 MATH3811 Page 3

i) Find the posterior density h(λ|X) of λ given X = (X1, X2, . . . , Xn).

Show that, like the prior, the posterior density is also a member of

the Gamma family.

ii) Find the Bayesian estimator of λ for quadratic error-loss with respect

to the prior τ(λ).

b) In a preliminary testing of a random number generator, the following ten

values were generated:

x1 = 0.621, x2 = 0.503, x3 = 0.203, x4 = 0.477, x5 = 0.710,

x6 = 0.581, x7 = 0.329, x8 = 0.480, x9 = 0.554, x10 = 0.382.

Carry out a Kolmogorov-Smirnov test to determine if the hypothesis of

a Uniform [0,1] distribution can be accepted. Use α = 0.05.

You can use the following extract of Table F for one-sample Kolmogorov

tests:

n 0.2 0.1 0.05 0.02 0.01

8 .358 .410 .454 .507 .542

9 .339 .387 .430 .480 .513

10 .323 .369 .409 .457 .489

(Each table entry is the value of the Kolmogorov statistic Dn for sample

of size n = 8, 9, 10 with the corresponding p-value given on the top row.)

3. [22 marks] Let X = (X1,X2, . . . ,Xn) be i.i.d. random variables, each with

a density

f(x, θ) =

1√2pixθe{−

1

2

[

ln(x)

θ

]2}, x > 0

0 elsewhere

where θ > 0 is a parameter. (This is called the log-normal density.)

a) Prove that the family L(X, θ) has a monotone likelihood ratio in T =∑n

i=1(lnXi)

2.

b) Argue that there is a uniformly most powerful (UMP) α−size test of the

hypothesis H0 : θ ≤ θ0 against H1 : θ > θ0 and exhibit its structure.

c) Using the density transformation formula (or otherwise) show that

Yi = lnXi

has a N(0, θ2) distribution.

d) Using c) (or otherwise), find the threshold constant in the test and hence

determine completely the uniformly most powerful α− size test ϕ∗ of

H0 : θ ≤ θ0 versus H1 : θ > θ0.

Calculate the power function Eθϕ

∗ and sketch a graph of it.

Please see over . . .

JUNE 2006 MATH3811 Page 4

4. [22 marks]

a) In an agricultural experiment, three blocks of similar size and location

have been selected. On each block, four varieties (A, B, C and D) of

potatoes have been grown on plots of equal size. The yields of potato

from the 12 plots have been registered in the table below:

Variety A Variety B Variety C Variety D

Block 1 68 67 71 77

Block 2 82 83 86 89

Block 3 56 59 64 60

Analyze these data using the Friedman test. At 10% level of significance,

decide if the hypothesis of equal yields for the four varieties can be ac-

cepted.

b) Two groups of students attempt a test with the following results:

Group 1: 91, 83, 76, 81

Group 2: 72, 61, 74, 63, 82

Using a (large sample) Wilcoxon test and α = 0.05, test the hypothesis

of no difference between the two groups.

Some useful formulae

1. Friedman’s Test:

F =

12l

K(K + 1)

K∑

i=1

(R¯i. − K + 1

2

)2 =

12

lK(K + 1)

K∑

i=1

R2i. − 3l(K + 1)

has χ2K−1 as a limiting distribution under the null hypothesis.

2. rth order statistic (r = 1, 2, . . . , n) of the sample of size n from a

distribution with a density fX(.) and a cdf FX(.) :

fX(r)(yr) =

n!

(r − 1)!(n− r)! [FX(yr)]

r−1[1− FX(yr)]n−rfX(yr)

Joint density of the couple (X(i), X(j)), 1 ≤ i < j ≤ n :

fX(i),X(j)(x, y) =

n!

(i− 1)!(j − 1− i)!(n− j)!fX(x)fX(y)[FX(x)]

i−1[FX(y)−FX(x)]j−1−i[1−FX(y)]n−j

for −∞ < x < y <∞.

3. Wilcoxon: Two independent samplesX1, X2, . . . , Xm and Y1, Y2, . . . Yn,

Wm+n =

∑m

i=1R(Xi). Then

Wm+n−m(m+n+1)/2√

mn(m+n+1)/12

→d N(0, 1).

4. Bayesian inference:

h(θ|X) = f(X|θ)τ(θ)

g(X)

, g(X) =

∫

Θ

f(X|θ)τ(θ)dθ.

5. Density transformation: Y = W (X) −→ fY (y) = fX(W−1(y))|d(W−1(y)dy |.

学霸联盟

SCHOOL OF MATHEMATICS

FINAL EXAMINATION

JUNE 2006

MATH3811

STATISTICAL INFERENCE

(1) TIME ALLOWED – 2 Hours

(2) TOTAL NUMBER OF QUESTIONS – 4

(3) ANSWER ALL QUESTIONS

(4) THE QUESTIONS ARE NOT OF EQUAL VALUE

(5) TOTAL NUMBER OF MARKS – 100

(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE

All answers must be written in ink. Except where they are expressly required pencils

may only be used for drawing, sketching or graphical work.

JUNE 2006 MATH3811 Page 2

1. [28 marks] Suppose a sample X = (X1, X2, . . . , Xn) from the density

f(x, θ) =

{

(1 + θ)xθ, x ∈ (0, 1),

0 elsewhere

is given where θ > −1 is the unknown parameter.

a) Calculate E(X1).

b) Find a minimal sufficient statistic T for θ. Give reasons for your answer.

c) Calculate the Fisher information about θ contained in the statistic T that

you found in b). Show that it is equal to n

(1+θ)2

.

d) Find the MLE of θ and also the MLE of h(θ) = 1

1+θ

.

e) Find the Cramer-Rao lower bound for the variance of an unbiased esti-

mator of h(θ) = 1

1+θ

.

f) Write down the score function. Find out if an UMVUE of h(θ) = 1

1+θ

exists and if it exists, write it down. Is there an unbiased estimator of

h(θ) = θ whose variance attains the Cramer-Rao bound? Explain your

answers.

g) State the asymptotic distribution of

√

n(hˆmle − h(θ)) for h(θ) = 11+θ .

Hint: The “delta method” implies that for any smooth function h(θ) :

√

n(h(θˆmle)− h(θ0))→d N(0, (∂h

∂θ

(θ0))

2I−1(θ0)),

I(θ) = E{ ∂

∂θ

[lnf(x, θ)]}2 = E{− ∂

2

∂θ2

[lnf(x, θ)]}.

2. [28 marks]

a) Let X = (X1, X2, . . . , Xn) be i.i.d. observations, each from a Poisson

distribution:

f(y|λ) = e(−λ) · λy/(y!), y = 0, 1, 2, . . . , λ > 0.

The prior on λ is believed to be Gamma(2,3).

(Note that, for any values of α > 0, β > 0, the density of the Gamma(α, β)

distribution is given by

τ(λ) =

{

1

βα·Γ(α) · λα−1 exp(−λ/β) , λ > 0;

0 else

and Γ(α) =

∫∞

0 e

−xxα−1dx is the Gamma function with the property

Γ(α+ 1) = αΓ(α).)

Please see over . . .

JUNE 2006 MATH3811 Page 3

i) Find the posterior density h(λ|X) of λ given X = (X1, X2, . . . , Xn).

Show that, like the prior, the posterior density is also a member of

the Gamma family.

ii) Find the Bayesian estimator of λ for quadratic error-loss with respect

to the prior τ(λ).

b) In a preliminary testing of a random number generator, the following ten

values were generated:

x1 = 0.621, x2 = 0.503, x3 = 0.203, x4 = 0.477, x5 = 0.710,

x6 = 0.581, x7 = 0.329, x8 = 0.480, x9 = 0.554, x10 = 0.382.

Carry out a Kolmogorov-Smirnov test to determine if the hypothesis of

a Uniform [0,1] distribution can be accepted. Use α = 0.05.

You can use the following extract of Table F for one-sample Kolmogorov

tests:

n 0.2 0.1 0.05 0.02 0.01

8 .358 .410 .454 .507 .542

9 .339 .387 .430 .480 .513

10 .323 .369 .409 .457 .489

(Each table entry is the value of the Kolmogorov statistic Dn for sample

of size n = 8, 9, 10 with the corresponding p-value given on the top row.)

3. [22 marks] Let X = (X1,X2, . . . ,Xn) be i.i.d. random variables, each with

a density

f(x, θ) =

1√2pixθe{−

1

2

[

ln(x)

θ

]2}, x > 0

0 elsewhere

where θ > 0 is a parameter. (This is called the log-normal density.)

a) Prove that the family L(X, θ) has a monotone likelihood ratio in T =∑n

i=1(lnXi)

2.

b) Argue that there is a uniformly most powerful (UMP) α−size test of the

hypothesis H0 : θ ≤ θ0 against H1 : θ > θ0 and exhibit its structure.

c) Using the density transformation formula (or otherwise) show that

Yi = lnXi

has a N(0, θ2) distribution.

d) Using c) (or otherwise), find the threshold constant in the test and hence

determine completely the uniformly most powerful α− size test ϕ∗ of

H0 : θ ≤ θ0 versus H1 : θ > θ0.

Calculate the power function Eθϕ

∗ and sketch a graph of it.

Please see over . . .

JUNE 2006 MATH3811 Page 4

4. [22 marks]

a) In an agricultural experiment, three blocks of similar size and location

have been selected. On each block, four varieties (A, B, C and D) of

potatoes have been grown on plots of equal size. The yields of potato

from the 12 plots have been registered in the table below:

Variety A Variety B Variety C Variety D

Block 1 68 67 71 77

Block 2 82 83 86 89

Block 3 56 59 64 60

Analyze these data using the Friedman test. At 10% level of significance,

decide if the hypothesis of equal yields for the four varieties can be ac-

cepted.

b) Two groups of students attempt a test with the following results:

Group 1: 91, 83, 76, 81

Group 2: 72, 61, 74, 63, 82

Using a (large sample) Wilcoxon test and α = 0.05, test the hypothesis

of no difference between the two groups.

Some useful formulae

1. Friedman’s Test:

F =

12l

K(K + 1)

K∑

i=1

(R¯i. − K + 1

2

)2 =

12

lK(K + 1)

K∑

i=1

R2i. − 3l(K + 1)

has χ2K−1 as a limiting distribution under the null hypothesis.

2. rth order statistic (r = 1, 2, . . . , n) of the sample of size n from a

distribution with a density fX(.) and a cdf FX(.) :

fX(r)(yr) =

n!

(r − 1)!(n− r)! [FX(yr)]

r−1[1− FX(yr)]n−rfX(yr)

Joint density of the couple (X(i), X(j)), 1 ≤ i < j ≤ n :

fX(i),X(j)(x, y) =

n!

(i− 1)!(j − 1− i)!(n− j)!fX(x)fX(y)[FX(x)]

i−1[FX(y)−FX(x)]j−1−i[1−FX(y)]n−j

for −∞ < x < y <∞.

3. Wilcoxon: Two independent samplesX1, X2, . . . , Xm and Y1, Y2, . . . Yn,

Wm+n =

∑m

i=1R(Xi). Then

Wm+n−m(m+n+1)/2√

mn(m+n+1)/12

→d N(0, 1).

4. Bayesian inference:

h(θ|X) = f(X|θ)τ(θ)

g(X)

, g(X) =

∫

Θ

f(X|θ)τ(θ)dθ.

5. Density transformation: Y = W (X) −→ fY (y) = fX(W−1(y))|d(W−1(y)dy |.

学霸联盟