xuebaunion@vip.163.com

3551 Trousdale Rkwy, University Park, Los Angeles, CA

留学生论文指导和课程辅导

无忧GPA：https://www.essaygpa.com

工作时间：全年无休-早上8点到凌晨3点

微信客服：xiaoxionga100

微信客服：ITCS521

统计代写-JUNE 2007

时间：2021-05-02

THE UNIVERSITY OF NEW SOUTH WALES

SCHOOL OF MATHEMATICS

FINAL EXAMINATION

JUNE 2007

MATH3911

HIGHER STATISTICAL INFERENCE

(1) TIME ALLOWED - 2 1/2 Hours

(2) TOTAL NUMBER OF QUESTIONS - 5

(3) ANSWER ALL QUESTIONS

(4) THE QUESTIONS ARE NOT OF EQUAL VALUE

(5) TOTAL NUMBER OF MARKS - 125

(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 2007 MATH3911 Page 2

1. [28 marks] Let X = (Xl, X 2, .. . ,Xn) (n 2: 2) be LLd. random variables

having Poisson distribution with density f(x, A) = e-:/x, x = 0,1,2, ...

a) Show that the product of indicators I{xl=o}(X)·I{xFo}(X) is an unbiased

estimator of the parameter T(A) = e-2A .

n

b) Given that T = L Xi is complete and minimal sufficient for A, derive

i=l

the UMVUE of T(A) = e-2A . (Hint: you may use part (a)).

c) Does the variance of the UMVUE in b) attain the Cramer-Rao bound

for the minimal variance of an unbiased estimator of T(A) = e-2A? Give

reasons for your answer.

d) Suppose now that for the same sample, the parameter of interest is h(A) =

~. Find the MLE of ~, state its asymptotic distribution and, using

this result, suggest a confidence interval for A that asymptotically has

a level 1 - Cl:. Explain why h(A) = ~ is called "variance stabilising

transformation" and why such a transformation it useful.

Hint: For any smooth function h(A) :

/Ti(h().,mle) - h(AO)) ~ N(O, (~~ (Ao))2 I-I (AO)),

8 82

I(A) = E{ 8A [lnf(x, A)]}2 = E{ - 8A2 [lnf(x, A)]}.

e) Let n = 6. Find the uniformly most powerful test of size Cl: = 0.1 for

testing Ho : A ::::: 0.25 against HI : A > 0.25. Justify your answer.

2. [24 marks]

a) A photocopier company kept records of the number of breakdowns per

year for 100 photocopiers, giving the results

Number of breakdowns 0 1 2 3 > 4

Frequency 8 20 37 24 11

Test at the 5% level that the above data comes from a Poisson (A = 2)

distribution.

b) Two companies manufacture white board pens. A sample of 10 pens from

one company and 8 pens from the other were tested until they were no

longer readable. The results (in hours) were:

Company 1 8.9 9.9 12 7.4 6.6 8.4 7.6 10.9 11.7 10.2

Company 2 9.1 13.3 8.2 11.1 10 11.6 10.5 8.7

Using the Wilcoxon Test for large samples, test at the 5% level the hy-

pothesis that there is no difference between the two company's pens.

c) The following table represents data on cortisol level of women in three

different groups: Group 1 delivered their babies by Caesarean section

Please see over ...

JUNE 2007 MATH3911 Page 3

,

Group 1 262 307 211 323 454 339 304 154 287 356

Group 2 465 501 455 355 468 362

Group 3 343 772 207 1048 838 687

before the onset of labor, Group 2 delivered by emergency Caesarean

during induced labor and group 3 experienced spontaneous labor.

Use the Kruskal-Wallis test at level 0.05 to test for equality of the average

cortisol level in the three groups.

3. [23 marks] Let X = (Xl, X 2 , .. . ,Xn ) be a sample of size n from a distribution

with density

f(x;O) = [~03/X4, x;::: 0> 0

.. . ., l u elsewhere

a) Prove that the density of X has a monotone likelihood ratio in X(1).

b) Find the cumulative distribution function of Z = X(1). Show that the

density is .

Jz(z;O) = {;::!:~ ,Z;::: 0> 0

o elsewhere

c) Find the uniformly most powerful a-size test rp* of Ho : 0 ::; 2 versus

H l : e> 2 and sketch a graph of Erp* as precisely as possible.

4. [28 marks] In a Bayesian framework, the probability of a defect eE (0,1) of

a certain piece of a machine's output appears to be varying on various days

and can be represented as a random variable with prior density

T(O) = 12e2 (1 - e), e E (0,1).

X. = { 1 if the piece is defective

t 0 if the piece is nondefective

Hence !(Xle) = e'£~=l Xi (1 _ e) n-'£~1 Xi.

On a certain day, n = 16 observations denoted X = (Xl, X 2 ,· .• ,X16 ) are

made where

a) Find the formula for the Bayesian estimator (with respect to quadratic

loss) of the probability e and calculate its estimated value if from the 16

pieces examined on this day, exactly three were defective.

Hint: You may use: B(a, {3) = J~ xa- l (1 - x)f3- l dx, B(a, {3) = ri~~~),

[(a) = Jooo exp( -x)xa-ldx, f(a + 1) = af(a).

b) Find the Bayesian estimator (with respect to quadratic loss ) of the pa-

rameter T/(e) = e(1- e).

c) Using the data of part (a) (3 defective out of 16 tested), test the hypoth-

esis Ho : e ::; 0.25 against H l : e> 0.25 using the Bayesian approach and

a 0-1 loss. Do you accept Ho?

(You may use: Jg.25 e5(1 - e)14de = 0.00000164).

Please see over ...

JUNE 2007 MATH3911 Page 4

5. [22 marks] Let X = (Xl, X 2 , . .. ,Xn ) be i.i.d., each with uniform [0,8) density

(8) 0).

a) Find COV(X(n_l), X(n)).

b) Find the density of the range R = X(n) - X(1). Using it, or otherwise,

show that ER = ~~~ 8 holds.

Some useful formulae

1) Wilcoxon: Two independent samples Xl, X 2,· . . , X m and Y1 , Y2, ... ,Yn ,

W = ,,~ R(X). Then W m +n -m(m+n+l)/2 has approximately a stan-

m+n L..~_l ~ y'mn(m+n+l)/12

dard normal distrihution.

2) The Kruskal-Wallis statistic for a K sample problem:

12 K _ n + 1 2 12 K RT.

H = L ni(Ri. - -) = L - - 3(n + 1)

n(n + 1) i=l 2 n(n + 1) i=l ni

has Xk-l as a limiting distribution under the null hypothesis.

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

distribution with a density !X(.) and a cdf Fx (.) :

The joint density of the pair (X(i)' X(j)), 1 :S i < j :S n is:

!x(i),x(j) (x, y) =

(i - l)!(j - ~!_ i)!(n _ j)! !X (x)!x (y)[Fx (xW-1 [Fx (y)-Fx (x)jJ-I-i[l-Fx (y)]n- j

for x < y.

4) Bayesian inference:

h(8IX) = !(~~i~(8), g(X) = le !(XI8)T(8)d8.

学霸联盟

SCHOOL OF MATHEMATICS

FINAL EXAMINATION

JUNE 2007

MATH3911

HIGHER STATISTICAL INFERENCE

(1) TIME ALLOWED - 2 1/2 Hours

(2) TOTAL NUMBER OF QUESTIONS - 5

(3) ANSWER ALL QUESTIONS

(4) THE QUESTIONS ARE NOT OF EQUAL VALUE

(5) TOTAL NUMBER OF MARKS - 125

(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 2007 MATH3911 Page 2

1. [28 marks] Let X = (Xl, X 2, .. . ,Xn) (n 2: 2) be LLd. random variables

having Poisson distribution with density f(x, A) = e-:/x, x = 0,1,2, ...

a) Show that the product of indicators I{xl=o}(X)·I{xFo}(X) is an unbiased

estimator of the parameter T(A) = e-2A .

n

b) Given that T = L Xi is complete and minimal sufficient for A, derive

i=l

the UMVUE of T(A) = e-2A . (Hint: you may use part (a)).

c) Does the variance of the UMVUE in b) attain the Cramer-Rao bound

for the minimal variance of an unbiased estimator of T(A) = e-2A? Give

reasons for your answer.

d) Suppose now that for the same sample, the parameter of interest is h(A) =

~. Find the MLE of ~, state its asymptotic distribution and, using

this result, suggest a confidence interval for A that asymptotically has

a level 1 - Cl:. Explain why h(A) = ~ is called "variance stabilising

transformation" and why such a transformation it useful.

Hint: For any smooth function h(A) :

/Ti(h().,mle) - h(AO)) ~ N(O, (~~ (Ao))2 I-I (AO)),

8 82

I(A) = E{ 8A [lnf(x, A)]}2 = E{ - 8A2 [lnf(x, A)]}.

e) Let n = 6. Find the uniformly most powerful test of size Cl: = 0.1 for

testing Ho : A ::::: 0.25 against HI : A > 0.25. Justify your answer.

2. [24 marks]

a) A photocopier company kept records of the number of breakdowns per

year for 100 photocopiers, giving the results

Number of breakdowns 0 1 2 3 > 4

Frequency 8 20 37 24 11

Test at the 5% level that the above data comes from a Poisson (A = 2)

distribution.

b) Two companies manufacture white board pens. A sample of 10 pens from

one company and 8 pens from the other were tested until they were no

longer readable. The results (in hours) were:

Company 1 8.9 9.9 12 7.4 6.6 8.4 7.6 10.9 11.7 10.2

Company 2 9.1 13.3 8.2 11.1 10 11.6 10.5 8.7

Using the Wilcoxon Test for large samples, test at the 5% level the hy-

pothesis that there is no difference between the two company's pens.

c) The following table represents data on cortisol level of women in three

different groups: Group 1 delivered their babies by Caesarean section

Please see over ...

JUNE 2007 MATH3911 Page 3

,

Group 1 262 307 211 323 454 339 304 154 287 356

Group 2 465 501 455 355 468 362

Group 3 343 772 207 1048 838 687

before the onset of labor, Group 2 delivered by emergency Caesarean

during induced labor and group 3 experienced spontaneous labor.

Use the Kruskal-Wallis test at level 0.05 to test for equality of the average

cortisol level in the three groups.

3. [23 marks] Let X = (Xl, X 2 , .. . ,Xn ) be a sample of size n from a distribution

with density

f(x;O) = [~03/X4, x;::: 0> 0

.. . ., l u elsewhere

a) Prove that the density of X has a monotone likelihood ratio in X(1).

b) Find the cumulative distribution function of Z = X(1). Show that the

density is .

Jz(z;O) = {;::!:~ ,Z;::: 0> 0

o elsewhere

c) Find the uniformly most powerful a-size test rp* of Ho : 0 ::; 2 versus

H l : e> 2 and sketch a graph of Erp* as precisely as possible.

4. [28 marks] In a Bayesian framework, the probability of a defect eE (0,1) of

a certain piece of a machine's output appears to be varying on various days

and can be represented as a random variable with prior density

T(O) = 12e2 (1 - e), e E (0,1).

X. = { 1 if the piece is defective

t 0 if the piece is nondefective

Hence !(Xle) = e'£~=l Xi (1 _ e) n-'£~1 Xi.

On a certain day, n = 16 observations denoted X = (Xl, X 2 ,· .• ,X16 ) are

made where

a) Find the formula for the Bayesian estimator (with respect to quadratic

loss) of the probability e and calculate its estimated value if from the 16

pieces examined on this day, exactly three were defective.

Hint: You may use: B(a, {3) = J~ xa- l (1 - x)f3- l dx, B(a, {3) = ri~~~),

[(a) = Jooo exp( -x)xa-ldx, f(a + 1) = af(a).

b) Find the Bayesian estimator (with respect to quadratic loss ) of the pa-

rameter T/(e) = e(1- e).

c) Using the data of part (a) (3 defective out of 16 tested), test the hypoth-

esis Ho : e ::; 0.25 against H l : e> 0.25 using the Bayesian approach and

a 0-1 loss. Do you accept Ho?

(You may use: Jg.25 e5(1 - e)14de = 0.00000164).

Please see over ...

JUNE 2007 MATH3911 Page 4

5. [22 marks] Let X = (Xl, X 2 , . .. ,Xn ) be i.i.d., each with uniform [0,8) density

(8) 0).

a) Find COV(X(n_l), X(n)).

b) Find the density of the range R = X(n) - X(1). Using it, or otherwise,

show that ER = ~~~ 8 holds.

Some useful formulae

1) Wilcoxon: Two independent samples Xl, X 2,· . . , X m and Y1 , Y2, ... ,Yn ,

W = ,,~ R(X). Then W m +n -m(m+n+l)/2 has approximately a stan-

m+n L..~_l ~ y'mn(m+n+l)/12

dard normal distrihution.

2) The Kruskal-Wallis statistic for a K sample problem:

12 K _ n + 1 2 12 K RT.

H = L ni(Ri. - -) = L - - 3(n + 1)

n(n + 1) i=l 2 n(n + 1) i=l ni

has Xk-l as a limiting distribution under the null hypothesis.

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

distribution with a density !X(.) and a cdf Fx (.) :

The joint density of the pair (X(i)' X(j)), 1 :S i < j :S n is:

!x(i),x(j) (x, y) =

(i - l)!(j - ~!_ i)!(n _ j)! !X (x)!x (y)[Fx (xW-1 [Fx (y)-Fx (x)jJ-I-i[l-Fx (y)]n- j

for x < y.

4) Bayesian inference:

h(8IX) = !(~~i~(8), g(X) = le !(XI8)T(8)d8.

学霸联盟