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程序代写案例-MATH5905

时间：2021-03-19

MATH5905 Term One 2021 Assignment One Statistical Inference

Problem One

During the cold winter months, the principle of the Winterville School District must decide

whether to call o↵ the next day’s school due to the snow conditions. If the principle fails to call

o↵ school and there is snow, there are serious concerns for children and teachers not showing

up for school and high rates of accidents. However, if the principle does call o↵ school and then

it does snow, the students undertake online classes and nothing is lost. On the other hand, if

the principle does call of school and it does not snow, the students must make-up the lost day

later in the year since it’s fine to go outside and students are unlikely to be able to focus in

the online class.

The principle decides that the costs of failing to close the school when there is snow is twice

the costs of closing the school when there is no snow. Therefore, the principle assigns two units

of loss to the first outcome and one to the second. If the principle closes the school when there

is snow or keeps it open when there is no snow, then no loss is incurred.

There are two local experts that provide the principle with independent and identically dis-

tributed weather forecasts. If there is snow, each expert will independently forecast snow with

probability 3/4 and no snow with probability 1/4. However, if there is to be no snow then each

expert independently predicts snow with probability 1/2. The principle will listen to both

forecasts and then make his decisions based on the data X which is the number of experts

forecasting snow.

a) There are two possible actions in the action space A = {a0, a1} where action a0 is to

keep the school open and action a1 is to close the school. There are two states of nature

⇥ = {✓0, ✓1} where ✓0 = 0 represents “no snow” and ✓1 = 1 represents “snow”. Define

the appropriate loss function L(✓, a) for this problem.

b) Compute the probability mass function (pmf) for X under both states of nature.

c) The complete list of all the non-randomized decisions rules D based on x is given by:

d1 d2 d3 d4 d5 d6 d7 d8

x = 0 a0 a1 a0 a1 a0 a1 a0 a1

x = 1 a0 a0 a1 a1 a0 a0 a1 a1

x = 2 a0 a0 a0 a0 a1 a1 a1 a1

For the set of non-randomized decision rules D compute the corresponding risk points.

d) Find the minimax rule(s) among the non-randomized rules in D.

e) Sketch the risk set of all randomized rules D generated by the set of rules in D. . You

might want to use R (or your favorite programming language) to make this sketch more

precise.

f) Suppose there are two decisions rules d and d0. The decision d strictly dominates d0 if

R(✓, d) R(✓, d0) for all values of ✓ and R(✓, d) < (✓, d0) for at least one value ✓. Hence,

given a choice between d and d0 we would always prefer to use d. Any decision rules

which is strictly dominated by another decisions rule (as d0 is in the above) is said to be

2

MATH5905 Term One 2021 Assignment One Statistical Inference

inadmissible. Correspondingly, if a decision rule d is not strictly dominated by any other

decision rule then it is admissible. Show on the risk plot the set of randomized decisions

rules that correspond to the principle’s admissible decision rules.

g) Find the risk point of the minimax rule in set of randomized decision rules D and deter-

mine its minimax risk. Compare the minmax risk from the minimax decision rule in D

and D.

h) Define the minimax rule in the set D in terms of rules in D.

i) For which prior on {✓1, ✓2} is the minimax rule in the set D also a Bayes rule?

j) Prior to listening to the forecasts, the principle believes there will be snow with probability

1/2. Find the Bayes rule and the Bayes risk with respect to this prior.

k) For a small positive ✏ = 0.1, illustrate on the risk set the risk points of all rules which

are ✏-minimax. Write down all the vertices that define the region.

Problem Two

Suppose that X is a binomial random variable with parameters n and ✓ where the number of

trials n is assumed to be known. Calculate the Bayes rule (based on a single observation of X)

for estimating ✓ when the prior distribution is the uniform distribution on [0, 1] and the loss

function is given by

L(✓, d) =

1

✓(1 ✓)(✓ d)

2.

Show that the rule you obtained is minimax.

Hint: The Beta function is given by

B(x, y) =

Z 1

0

tx1(1 t)y1dt

and the following relation holds:

B(x+ 1, y)

B(x, y)

=

x

x+ y

.

Problem Three

Suppose X = (X1, . . . , Xn) are i.i.d. Exponential(✓) with density

f(x|✓) = ✓e✓x, x > 0, ✓ > 0

and let ✓ have a Gamma(↵,) prior distribution with density

⌧(✓) =

1

(↵)↵

✓↵1e✓/ , ↵, > 0, ✓ > 0.

a) Find the posterior distribution for h(✓|X). Use the notation s =Pni=1Xi.

3

MATH5905 Term One 2021 Assignment One Statistical Inference

b) Hence or otherwise determine the Bayes estimator of ✓ with respect to the quadratic loss

function L(a, ✓) = (a ✓)2.

c) Suppose the following ten observations were observed:

0.12, 0.28, 0.43, 0.34, 0.47, 0.67, 0.82, 0.12, 0.30, 0.45

Two actions a0 (accept H0) and a1 (reject H0) are possible and the losses when using

these actions are given by:

L(✓, a0) =

(

0 if ✓ 2 ⇥0

2 if ✓ 2 ⇥1

and L(✓, a1) =

(

1 if ✓ 2 ⇥0

0 if ✓ 2 ⇥1

Using the loss function above and the parameters ↵ = 2 and = 1 for the prior, what

is your decision when testing H0 : ✓ 2.5 versus H1 : ✓ > 2.5. You may use the pgamma

function in R or another numerical integration routine from your favorite programming

package to answer this question.

Problem Four

Determine the form of the Bayes decision rule in an estimation problem with a one-dimensional

parameter ✓ 2 R1 and loss function

L

✓, d

=

(

↵

✓ d if d ✓,

d ✓ if d > ✓,

where ↵ and are known positive constants.

4

学霸联盟

Problem One

During the cold winter months, the principle of the Winterville School District must decide

whether to call o↵ the next day’s school due to the snow conditions. If the principle fails to call

o↵ school and there is snow, there are serious concerns for children and teachers not showing

up for school and high rates of accidents. However, if the principle does call o↵ school and then

it does snow, the students undertake online classes and nothing is lost. On the other hand, if

the principle does call of school and it does not snow, the students must make-up the lost day

later in the year since it’s fine to go outside and students are unlikely to be able to focus in

the online class.

The principle decides that the costs of failing to close the school when there is snow is twice

the costs of closing the school when there is no snow. Therefore, the principle assigns two units

of loss to the first outcome and one to the second. If the principle closes the school when there

is snow or keeps it open when there is no snow, then no loss is incurred.

There are two local experts that provide the principle with independent and identically dis-

tributed weather forecasts. If there is snow, each expert will independently forecast snow with

probability 3/4 and no snow with probability 1/4. However, if there is to be no snow then each

expert independently predicts snow with probability 1/2. The principle will listen to both

forecasts and then make his decisions based on the data X which is the number of experts

forecasting snow.

a) There are two possible actions in the action space A = {a0, a1} where action a0 is to

keep the school open and action a1 is to close the school. There are two states of nature

⇥ = {✓0, ✓1} where ✓0 = 0 represents “no snow” and ✓1 = 1 represents “snow”. Define

the appropriate loss function L(✓, a) for this problem.

b) Compute the probability mass function (pmf) for X under both states of nature.

c) The complete list of all the non-randomized decisions rules D based on x is given by:

d1 d2 d3 d4 d5 d6 d7 d8

x = 0 a0 a1 a0 a1 a0 a1 a0 a1

x = 1 a0 a0 a1 a1 a0 a0 a1 a1

x = 2 a0 a0 a0 a0 a1 a1 a1 a1

For the set of non-randomized decision rules D compute the corresponding risk points.

d) Find the minimax rule(s) among the non-randomized rules in D.

e) Sketch the risk set of all randomized rules D generated by the set of rules in D. . You

might want to use R (or your favorite programming language) to make this sketch more

precise.

f) Suppose there are two decisions rules d and d0. The decision d strictly dominates d0 if

R(✓, d) R(✓, d0) for all values of ✓ and R(✓, d) < (✓, d0) for at least one value ✓. Hence,

given a choice between d and d0 we would always prefer to use d. Any decision rules

which is strictly dominated by another decisions rule (as d0 is in the above) is said to be

2

MATH5905 Term One 2021 Assignment One Statistical Inference

inadmissible. Correspondingly, if a decision rule d is not strictly dominated by any other

decision rule then it is admissible. Show on the risk plot the set of randomized decisions

rules that correspond to the principle’s admissible decision rules.

g) Find the risk point of the minimax rule in set of randomized decision rules D and deter-

mine its minimax risk. Compare the minmax risk from the minimax decision rule in D

and D.

h) Define the minimax rule in the set D in terms of rules in D.

i) For which prior on {✓1, ✓2} is the minimax rule in the set D also a Bayes rule?

j) Prior to listening to the forecasts, the principle believes there will be snow with probability

1/2. Find the Bayes rule and the Bayes risk with respect to this prior.

k) For a small positive ✏ = 0.1, illustrate on the risk set the risk points of all rules which

are ✏-minimax. Write down all the vertices that define the region.

Problem Two

Suppose that X is a binomial random variable with parameters n and ✓ where the number of

trials n is assumed to be known. Calculate the Bayes rule (based on a single observation of X)

for estimating ✓ when the prior distribution is the uniform distribution on [0, 1] and the loss

function is given by

L(✓, d) =

1

✓(1 ✓)(✓ d)

2.

Show that the rule you obtained is minimax.

Hint: The Beta function is given by

B(x, y) =

Z 1

0

tx1(1 t)y1dt

and the following relation holds:

B(x+ 1, y)

B(x, y)

=

x

x+ y

.

Problem Three

Suppose X = (X1, . . . , Xn) are i.i.d. Exponential(✓) with density

f(x|✓) = ✓e✓x, x > 0, ✓ > 0

and let ✓ have a Gamma(↵,) prior distribution with density

⌧(✓) =

1

(↵)↵

✓↵1e✓/ , ↵, > 0, ✓ > 0.

a) Find the posterior distribution for h(✓|X). Use the notation s =Pni=1Xi.

3

MATH5905 Term One 2021 Assignment One Statistical Inference

b) Hence or otherwise determine the Bayes estimator of ✓ with respect to the quadratic loss

function L(a, ✓) = (a ✓)2.

c) Suppose the following ten observations were observed:

0.12, 0.28, 0.43, 0.34, 0.47, 0.67, 0.82, 0.12, 0.30, 0.45

Two actions a0 (accept H0) and a1 (reject H0) are possible and the losses when using

these actions are given by:

L(✓, a0) =

(

0 if ✓ 2 ⇥0

2 if ✓ 2 ⇥1

and L(✓, a1) =

(

1 if ✓ 2 ⇥0

0 if ✓ 2 ⇥1

Using the loss function above and the parameters ↵ = 2 and = 1 for the prior, what

is your decision when testing H0 : ✓ 2.5 versus H1 : ✓ > 2.5. You may use the pgamma

function in R or another numerical integration routine from your favorite programming

package to answer this question.

Problem Four

Determine the form of the Bayes decision rule in an estimation problem with a one-dimensional

parameter ✓ 2 R1 and loss function

L

✓, d

=

(

↵

✓ d if d ✓,

d ✓ if d > ✓,

where ↵ and are known positive constants.

4

学霸联盟