Mathematics Ph.D. Qualifying ExaminationStat 52800 Mathematical Statistics, January of 2020

NOTE: Answer all questions completely. Justify every step. Time allowed: 3 hours.

1. Let X1, . . . , Xn be a random sample from the normal distribution, N (µ, σ2), with mean

µ and finite variance σ2 > 0. Based on the likelihood ratio criterion, construct a test

for the hypotheses H0 : σ2 = σ′2 specified, µ unspecified against H1 : σ2 6= σ

′2, µ

unspecified.

2. Let X1, . . . , Xn be independent and identically distributed (i.i.d.) random variables

with the uniform distribution U(0, θ) over (0, θ) for θ > 0.

(a) Show that θn = max{X1. . . . , Xn} is the maximum likelihood estimator (MLE) of

θ. Give the limiting distribution of Zn = n(θ − θn).

(b) For a real number a, let θn(a) = aθn. Find the value a of a such that θn(a) is an

unbiased estimator of θ. Derive the limiting distribution of θn(a).

(c) Let h(a) = E{(θn(a)− θ)2}. Find the value a of a which minimizes h(a). Derive

the limiting distribution of θn(a).

(d) Find the value of c so that θn = cXn is an unbiased estimator of θ. Give the

limiting distribution for θn.

(e) Compare these estimators of θ: θn, θn(a), θn(a) and θn. Which estimator(s) do

you suggest to use? Why?

3. Suppose X1, . . . , Xn are i.i.d. with the normal distribution, N (0, σ2), with mean zero

and finite variance σ2 > 0. Answer the following questions.

(a) Show that the distribution belongs to the exponential family.

(b) Give the complete sufficient statistic for σ.

(c) Find the constant C so that σ = C

n

∑n

i=1|Xi| is an unbiased estimator of σ. Give

the asymptotic distribution of σ.

(d) Find the MLE σ of σ. Give the asymptotic distribution of σ.

(e) Calculate the relative efficiency of the above two estimators.

(f) Derive the uniformly most powerful test (UMP) of size α for testing H0 : σ = 1

versus σ < 1.

4. Let (X1, Y1), (X2, Y2), · · ·, (Xn, Yn) be a random sample from a bivariate normal distri-

bution with marginal means µ1, µ2, marginal variances σ2

1= σ2

2= σ2, and correlation

coefficient ρ = 1/2, where µ1, µ2 and σ2 > 0 are unknown real numbers. Find the like-

lihood ratio test Λ for testing H0 : µ1 = µ2 = 0, σ2 unknown against all alternatives.

5. Let X1, · · · , Xn denote a random sample of size n ≥ 2 from a distribution with pdf

f(x; θ) = θxθ−1, x > 0 and zero elsewhere, where θ > 0 is a parameter. Answer the

following questions.

(a) Find the MLE θ of θ. Give the asymptotic distribution.

(b) Find c such that θ = cθ is an unbiased estimator of θ. Give the asymptotic

distribution.

(c) Are θ and θ asymptotically efficient? Justify.

(d) Give the MLE of h(θ) for a measurable function h and its asymptotic distribution

when h is continuously differentiable.

2

Mathematics Ph.D. Qualifying ExaminationStat 52800 Math Statistics, Fall 2019

NOTE: Answers all questions completely. Justify every step. Time allowed: 3 hours.

1. Let X1, ..., Xn be independent and identically distributed (iid) with pdf f(x; θ) =

1/(3θ),−θ < x < 2θ, zero elsewhere, where θ > 0.

(a) Find the MLE θ of θ, and give the limiting distribution.

(b) Is θ a sufficient statistic for θ? Justify.

(c) Find the MVUE θ of θ, and give the limiting distribution.

(d) Find the moment estimate θ of θ, and give the limiting distribution.

(e) Which estimate, θ, or θ, or θ, do you suggest to use? Why?

2. Let X have a pmf of the form f(x; θ) = θx(1 − θ)1−x, x = 0, 1, zero elsewhere, where

0 < θ < 1. Consider testing the hypotheses:

H0 : θ = 1/20 vs H1 : θ > 1/20.

Use the Central Limit Theorem to determine the sample size n of a random sample so

that a uniformly most powerful test of H0 against H1 has a significance level α = 0.05

and power 0.99 at θ = 1/10.

3. Let X1, . . . , Xn be a random sample from the exponential distribution with pdf

f(x; θ) = (1/θ) exp(−x/θ), x > 0,

where θ > 0. Consider testing the hypotheses

H0 : θ = θ0, vs θ 6= θ0,

where θ0 is some pre-specified number.

(a) Derive the likelihood ratio test (LRT), and give the rejection region.

(b) Show that the LRT rejects the null hypothesis if and only if W/θ0 ≤ c1 or W/θ0 ≥

c2 for some constants c1, c2, where W = X1 + . . .+Xn.

(c) Show under the null hypothesis H0, 2W/θ0 has chisquare distribution with 2n

degrees of freedom, i.e.,

2W/θ0 ∼ χ2(2n).

(d) For θ0 = 3 and n = 5, give the equations for c1, c2 so that the test that rejects H0

when W ≤ c1 and W ≥ c2 has significance level α = 0.10.

4. Let Xn denote the mean of a random sample of size n from a distribution that has pdf

f(x) = exp(−x), 0 < x < ∞, zero elsewhere.

(a) Show that the mgf MYn(t) of Yn =

√n(Xn − 1) is

MYn(t) = (et/

√

n − (t/√n)et/

√

n)−n, t <√n.

(b) Find the limiting distribution of Yn.

(c) Use the ∆-method to find the limiting distribution of Zn =√n((Xn)

1/2 − 1).

5. Let X1, . . . , Xn be a random sample from normal distribution with mean µ variance

σ2 > 0. Let X be the sample mean and S2 be the sample variance, i.e.,

X =1

n

n∑

i=1

Xi, S2 =1

n− 1

n∑

i=1

(Xi − X)2, n ≥ 2.

Show that X and S2 are independent.

2

Mathematics Ph.D. Qualifying ExaminationStat 52800 Math Statistics, Fall 2018

NOTE: Answers all questions completely. Justify every step. Time allowed: 3 hours.

1. We are interested in testing about the fairness of a coin. Let θ ∈ (0, 1) denote the

probability of the coin landing on heads, and consider testing H0 : θ = 0.5 versus

H1 : θ > 0.5. We toss the coin until we see the first head. Let X denote the number

of tosses. Find the uniformly most powerful test for the above hypotheses at the level

0 < α < 1 of significance.

2. Let X1, . . . , Xn be a random sample from N(µ, σ2).

(a) Find the maximum likelihood estimate (MLE) b of b determined by P (X1 ≤ b) =

0.95.

(b) Give the limiting distribution of b.

3. Let Y1 < Y2 < Y3 be the order statistics of a random sample of size 3 from the density

f(x; θ1, θ2) =1

θ2exp(−x−θ1

θ2), x > θ1, and zero elsewhere. Here θ2 > 0. Find the joint

pdf of Z1 = Y1, Z2 = Y2 and Z3 = Y1 + Y2 + Y3. Show Z1 and Z3 are jointly sufficient

for θ1, θ2.

4. Consider the pdf

f(x) =

{λ exp(−λx), x > 0

0, elsewhere

where λ > 0. Generate an observation from this pdf using

(a) The inverse distribution functiom method.

(b) The accept-reject algorithm.

5. Let Y1 < Y2 < . . . < Yn be the order statistics of a random sample of size n from a

distribution that has pdf f(x; θ) = θ exp(−θx), 0 < x < ∞, zero elsewhere. Show that

the ratio R = nY1/∑

n

i=1Yi and its denominator S =

∑n

i=1Yi are independent.

6. Let X1, . . . , Xn be a random sample from this distribution with pdf

f(x; θ) =x3

6θ4exp(−

x

θ), x > 0, θ > 0.

(a) Find a complete sufficient statistic Y for θ

(b) Find the MLE θ of θ.

(c) Find a MVUE θ of θ. Is it unique?

7. Let Z∗ be drawn at random from the discrete distribution which has probability mass

1/n at each point zi = xi − x + µ0, that is, P (Z∗ = zi) = 1/n for i = 1, . . . , n. Here

x1, . . . , xn are observations and µ0 is a constant. Determine E(Z∗) and V ar(Z∗).

Mathematics Ph.D. Qualifying ExaminationStat 52800 Probability, January 2018

NOTE: Answers all questions completely. Justify every step. Time allowed: 3 hours.

1. Let X1, . . . , Xn be a random sample from a normal distribution N(θ, 1). Answer the

following questions.

(1) Consider the hypotheses

H0 : θ = θ0 against H1 : θ > θ0.

Derive a uniformly most powerful (UMP) test at the level α ∈ (0, 1) of significance.

A random sample of size n = 100 resulted in the sample mean xn = 10.160, do you

reject H0 when θ0 = 10 at the level α = .05 of significance?

(2) Derive the likelihood ratio test at the level α ∈ (0, 1) of significance for testing

H0 : θ = θ0 against H1 : θ = θ0,

where θ0 is specified. (Give the rejection rule).

(3) Is the test in (2) an unbiased test of H0 against H1? Justify your conclusion.

(4) Is the test in (2) a uniformly most powerful test of H0 against H1? Justify your

conclusion.

2. Let X1, . . . , Xn denote a random sample from N (0, θ), where the variance θ is an

unknown positive number. Show that there exists a uniformly most powerful test with

significance level α for testing the simple hypothesis H0 : θ = θ′, where θ′ is a fixed

positive number, against the alternative composite hypothesis H1 : θ > θ′.

3. Show that Y = |X| is a complete sufficient statistic for θ > 0, where X has the pdf

f(x; θ) = 1/(2θ), for −θ < x < θ, zero elsewhere. Show that Y = |X| and Z = sgn(X)

(i.e. Z = 1 if X > 0 and Z = −1 if X < 0) are independent.

4. Suppose f(x; θ) is twice continuously differentiable w.r.t. θ ∈ Θ with the score function

S(θ; x) = ∂∂θ

log f(x; θ), the Fisher information I(θ) = E([S(θ;X)]2), and the Hessian

function H(θ) = E( ∂2

∂2θlog f(X; θ)) (provided that the latter two expected values are

finite). Derive

E(S(θ;X)) = 0, I(θ) = −H(θ).

In deriving the above two equalities, what assumptions are needed (list as many as you

think are needed)?

5. Let X1, . . . , Xn be i.i.d. with E(X1) = µ and V ar(X1) = σ2 ∈ (0,∞). Let X =

(X1 + . . .+Xn)/n and S2n =

∑ni=1(Xi − X)2/(n− 1). Show Tn =

√n(X−µ)Sn

converges in

distribution to the standard normal.

6. Suppose X1, . . . , Xn are independent and identically distributed with density f(x; θ) =

θxθ−1 with 0 < x < 1 and θ > 0 unknown.

(1) Find the maximum likelihood estimator of θ.

(2) Assuming the asymptotic theory for the MLE applies, give the asymptotic standard

deviation for your estimator and use it to construct a 90% confidence interval for

θ.

(3) Describe the assumptions needed to justify the work required in (2). Without

bogging down in this question, discuss how to check those assumptions apply in

this situation.

2

Mathematics Qualifying Examination –January 2016

STAT 52800 - Mathematical Statistics

NOTE: Give answers to all questions and completely justify your derivations and steps.

A calculator and statistical tables (normal, t and chi-square) are allowed. Time: 3 hours.

1. Suppose that X1,X2, . . . ,Xn is a random sample from the Poisson distribution

Pois(λ), (with λ > 0).

a) Find the best unbiased estimator of ψ(λ) = λ exp(−λ).

b) Prove that there is no unbiased estimator for ψ(λ) = 1/λ .

2. Let X1,X2, . . . ,Xn be a random sample from the normal N (µ, σ2) distribution,

where µ and σ2 are unknown parameters. Let ηα denote that (1−α)×100 percentile

of that distribution.

a) Find the MLE ηn,α of ηα and show it is a biased estimator.

b) Find the limiting distribution of√n(ηn,α − ηα).

3. Let X1,X2, . . . ,Xn and Y1, Y2, . . . , Ym be two independent random samples from

f(x |λ1 , γ) and f(x |λ2 , γ), where, f(x |λ, γ) = λ e−λ(x−γ)I[x > γ] , with λ1 > 0, λ2 >

0 and γ ∈ R .

a) Assuming that λ1 and λ2 are both known, find the MLE γn,m of γ and find it

distribution.

b) Is the MLE γn,m an unbiased estimator of γ ? Prove its consistency as n → ∞and m→ ∞ .

4. Refer to problem 3 above and assume that γ = γ0 ≡ 0 is known and that λ1 and λ2

are two unknown parameters.

a) Construct an appropriate α -level likelihood ratio test of H0 : λ1 = λ2 against

H1 : λ1 6= λ2 and show that the test can be expressed in terms of Xn/Ym .

b) Derive the power function of this test.

5. Let Xij , i = 1, . . . , a j = 1, . . . , n be independent random variables such that Xij ∼N (µi, σ

2), where µi = µ + αi, i = 1, . . . a , with α1, . . . , αa satisfying∑

i αi = 0.

Here, µ, σ and α1, . . . , αa are all unknown parameters.

Define X·· =∑

i,j Xij/na, Xi· =∑

j Xij/n and let

Q1 =∑

i,j

(Xij − X··)2, Q2 =

∑

i

(Xi − X··)2, Q3 =

∑

i,j

(Xij − Xi·)2.

1

a) Write down the estimates of all parameters involved in the model.

b) Find the distribution of Q`/σ2 , ` = 1, 2, 3. If any non-centrality parameters are

involved, give them explicitly.

c) Let MS2 = Q2/d2 and MS3 = Q3/d3 , where d` denote the degrees of freedom

associated with Q`, ` = 2, 3. Find the exact distribution of the ratio F ∗ =

MS2/MS3 , under the hypothesis H0 : α1 = α2 = · · · = αa .

6. State Basu’s Theorem and give (or outline) its proof. Give a concrete example for its

application.

2

Mathematics Qualifying Examination –January 2015

STAT 52800 - Mathematical Statistics

NOTE: Answer all questions completely and justify your derivations and steps. A calcu-

lator and statistical tables (normal, t and chi-square) are allowed. Time: 3 hours.

1. Suppose that the random vector X˜

= (X1, . . . ,Xn)′ has the multi-normal distribu-

tion Nn(µ˜, σ2In), with µ

˜= (µ1, . . . , µn)′ and In is the n × n identity matrix. Let

X = 1n

1˜′

nX˜

, with 1˜

n = (1, 1, . . . , 1)′ , be the usual average.

a) Find the exact distribution of the quadratic form, Q = n(X)2 .

b) Evaluate E(Q) when n = 10, σ2 = 4 and µi = 2,∀i .

2. Consider the SLR model Yi = β0+β1xi+εi , with εi ∼ N (0, σ2), i.i.d. , i = 1, 2, . . . , n .

Let η0 ≡ E(Y |x = x0) = β0 + β1x0 , be the mean response of Y evaluated at some

fixed value x0 of x .

a) Derive the LSE η0 of η0 ?

b) Calculate the mean and variance of this estimator, η0 ?

c) Obtain a (1 − α) × 100% confidence interval for η0 .

3. Let X1,X2, . . . ,Xn be a random sample from a continuous distribution whose p.d.f.

is

f(x |λ, γ) = λ e−λ(x−γ) for x > γ with λ > 0 and γ ∈ R.

a) Find the minimal sufficient statistic for θ ≡ (λ, γ).

b) Assuming that λ = λ0 is known, find the MLE for γ and obtain its pdf. Is it

complete (prove or disprove)?

c) Assuming that γ = γ0 is known, find the MLE for λ and obtain its pdf. Is it

complete (prove or disprove)?

d) Find the (joint) MLE θn ≡ (λn, γn) for θ = (λ, γ).

e) Prove that λn and γn of part d) are independent r.v.’s.

f) Find the MLE, ψn , of the reliability

ψ(λ, γ) = Pr(X ≥ t0 | λ, γ) ≡

∫∞

t0

f(x |λ, γ)dx,

at some known t0 > γ .

1

4. Refer to problem 3 above.

a) Construct an appropriate α -level likelihood ratio (LR) test of H0 : λ ≤ λ0 against

H1 : λ > λ0 (when γ is known, say γ = 0).

b) Find an expression for the power function of this test.

c) Obtain an explicit expression for the critical value of LR test in terms of the

appropriate quantile of a well known distribution.

5. Refer to problem 3 above and assume now that λ = 1. Let ξ(γ) = γ2 . Use results

you obtained in problem 3 to derive an explicit expression for the UMVUE ξn of ξ .

6. Suppose we have four identical coins and we would like to test H0 : p ≤ 1/2 versus

H1 : p > 1/2, where p ∈ (0, 1) is the unknown probability of a Head for any one of

the coins. We decide to perform the following experiment: Each one of the coins will

be tossed repeatedly until the first Head occurs. Let Xi denote the number of Tails

counted until the first Head occurs for the ith coin, i = 1, 2, 3, 4.

a) Construct, based on the data, X1, . . . ,X4 , a size α = 3/16 UMP test of H0

versus H1 .

b) What is the power of the above UMP test at p = 1/4 and p = 3/4?

2

Mathematics Qualifying Examination –August 2014STAT 52800 - Mathematical Statistics

NOTE: Answer all questions completely and justify your derivations and steps. A calcu-lator and statistical tables (normal, t and chi-square) are allowed. Time: 3 hours.

1. Let Y1, Y2, . . . , Yn be a random sample from a Bernoulli distribution with parameterθ , where θ is restricted to the interval Θ ≡ (0, 3/4] . Find the MLE of ν = θ(1 − θ).

2. Let X1,X2, . . . ,Xn be i.i.d. observations from the gamma distribution

f(x|λ) =Xα−1e−x/λ

Γ(α)λα, 0 < x < ∞,

with known shape parameter α > 0 and unknown scale parameter λ .a) What is the sufficient statistics for λ? Is it minimal?b) Find the maximum likelihood estimator (MLE) of the reliability at a known t0 > 0

given γ(λ) =∫ ∞

t0f(t|λ)dt .

c) Find the Cramer-Rao lower bound for the variance of an unbiased estimator ofγ(λ) based on X1, . . . ,Xn or a suitable transformation thereof.

d) Is the MLE of γ(λ) consistent and asymptotically efficient? Support your asser-tions.

3. Let X1,X2, . . . ,Xn be a random sample from N (0, σ2).a) Find the UMVUE for σ2 .

b) Show that your answer to part a above is statistically independent ofX(1)

X(2), the

ratio of the first two order statistics of the given sample.

4. Suppose that the independent random variables Y1, Y2, . . . , Yn satisfy

Yi = βxi + εi, for, i = 1, 2, . . . , n,

where x1, . . . , xn are some known constants, β is an unknown regression parameter,and εi ∼ N (0, σ2), are iid and σ2 is a known constant.a) Construct the Likelihood Ratio Test (LRT) of H0 : β = 0, versus H1 : β 6= 0.

b) Suppose that n = 100 and∑100

1 xi = 10 and σ2 = 5. Find the exact rejectionregion of a size α = 0.05 LRT you constructed in part (a).

5. Suppose we have four identical coins and we would like to test H0 : p = 0.5, versusH1 : p > 0.5, where p is the unknown probability that any one of these coinscomes up ’heads’ when it is flipped. We decided to perform the following experiment.Each coin will be flipped repeatedly until the first ’head’ occurs. Let Xi denote thenumber of ’tails’ that occur before the first ’head’ occurs when the ith coin is flipped,i = 1, 2, 3, 4. Use these data, X1,X2,X3,X4 to construct a Uniformly Most Powerful(UMP) test of size α = 3/16 of H0 versus H1 .

1

6. Consider the problem of testing, based on a sample of size 1, of the hypothesis H0:X ∼N (0, 1) against the alternative hypothesis H1:X ∼ εN (0, 1)+ (1− ε)C(0, 1) for someunknown ε > 0, where C(0, 1) stands for the standard Cauchy distribution withdensity 1

π(1+x2) , −∞ < x < ∞ .

a) Are H0 and H1 both simple hypotheses?b) Show that a UMP test of level .01 exists for this problem and that in fact it

coincides with the usual test for testing that the mean θ of a normal distributionwith variance 1 equals zero (which is to reject H0 if |x| ≥ 2.58).

7. Let X and Y two Bernoulli random variables such that P (X = 1) = P (Y = 1) =p, p ∈ (0, 1) and P (X = Y = 1) = θ .a) Prove that θ ≤ p .b) Calculate the (simple) correlation between X and Y , namely, ρ

XY≡ Cor(X,Y ).

c) For X as above, let Mp ≡ {g : Ep(g(X)) = 0 and V arp(g(X)) = 1 } . Find allmembers of the class of functions Mp . (Hint: There are two such members inMp .)

d) For X and Y as above, define the maximal correlation over Mp as

ρM

≡ supg,u∈Mp

Cor(g(X), u(Y )).

Calculate ρM and compare it to ρXY

.

2

Mathematics Ph.D. Qualifying Examination January 2013STAT 52800 Mathematical Statistics

NOTE: Answer all four questions completely. Justify every step. A calculator and somestatistical tables (normal, t and chi-square) are allowed. Time allowed: 3 hours.

1. Suppose that X1, . . . , Xn is a random sample from the probability density function

f(x|θ) =2x

θe−x2/θ, 0 < x <∞,

where θ > 0 is an unknown parameter.

(a) Find the method of moments estimator (MME) of θ and the maximum likeli-hood estimator (MLE) of θ.

(b) Are the MME and MLE above (i) consistent? (ii) unbiased? Give reasons.

(c) Which estimator, MME or MLE, do you favor using and why?

(d) Based on the asymptotic distribution of the MLE for θ, construct a 95% con-fidence interval for θ.

2. Let Y1 < Y2 < . . . < Yn be the order statistics of a random sample from alognormal(θ, 1) distribution, where θ > 0 is an unknown parameter.

(a) Find the minimum variance unbiased estimator (MVUE) of θ or of ψ = e−θ.The choice is yours.

(b) Find the maximum likelihood estimator (MLE) of ψ = e−θ and of θ.

(c) If ψn is either the MVUE or the MLE of ψ, show that, nr(ψn − ψ) → 0 inprobability as n→ ∞, for any r ∈ [0, 0.5).

(d) Derive an unbiased estimator ηn of η = Φ(−θ), where Φ(·) is the cumulativedistribution function of the standard normal variable. Starting from this ηn,how will you derive the MVUE of η?

3. Let X1, . . . , Xn be IID beta(1, ν−1), where 0 < ν < ∞ is an unknown parameter.Note that instead of working with X, you may prefer to work with Y = − ln(1−X).

(a) What is the maximum likelihood estimator (MLE) of ν? Is the MLE of ν alsoa complete sufficient statistic for ν?

(b) What is the best 5% level critical region for testing H0 : ν ≤ 5 versus H1 : ν >5? In what sense is it the best?

(c) How big a sample is needed so that the above test will attain a 10% probabilityof type II error at ν = 6?

1

(d) Develop a sequential probability ratio test for testing H0 : ν = 5 versus H2 :ν = 6 that will attain a 5% probability of type I error and a 10% probabilityof type II error.

4. (a) The table below gives the number of days spent in the ICU by all patientsadmitted during the week of December 23–29, 2012. This information is gatheredafter all such patients have been discharged from the ICU.

Table 1: Duration of Stay in ICU

# Days 1 2 3 4 5 total# Patients 20 30 12 9 9 80

Construct a 95% confidence interval for μ, the average number of days spent in theICU by each patient.

(b) Now suppose that we consider a different survey design. The surveyor visits theICU at noon of each Sunday in December 2012 and makes a list of all patients inthe ICU. Later on she collects the number of days these patients spent in the ICU.

Table 2: Duration of Stay in ICU Using New Survey

# Days 1 2 3 4 5 total# Patients 10 30 18 19 23 100

Based on the data in Table 2, construct a 95% confidence interval for μ.

2

Mathematics Ph.D. Qualifying Examination August 2012STAT 52800 Mathematical Statistics

NOTE: Answer all four questions completely. Justify every step. A calculator and somestatistical tables (normal, t and chi-square) are allowed. Time allowed: 3 hours.

1. Suppose that X1, . . . , Xn is a random sample from the probability density function

f(x|θ) =r xr−1

θe−xr/θ, 0 < x <∞,

where r > 1 is a known constant and θ > 0 is an unknown parameter.

(a) Find an estimator of θ by the method of moments.

(b) Find the MLE of θ.

(c) Are the estimators in (a) and (b) consistent? (Show why or why not.)

(d) Which estimator, (a) or (b), would you favor using and why?

(e) Based on the MLE for θ, find an unbiased estimator of θ.

(f) Based on the asymptotic distribution of the MLE for θ, construct a 95% con-fidence interval for θ.

2. Let X1, . . . , Xn be a random sample from a Poisson(λ) distribution, where λ > 0 isthe mean parameter.

(a) Find the uniformly minimum variance unbiased estimator (UMVUE) ψn andthe maximum likelihood estimator (MLE) ψn of ψ = e−λ.

(b) Compare the UMVUE and the MLE of ψ. Are the two similar or quite differ-ent? Explain.

(c) Show that, for any r ∈ [0, 0.5), the UMVUE satisfies nr(ψn − ψ) → 0 inprobability as n→ ∞.

(d) Derive the UMVUE ηn of η = P{Xi ≤ 1}. Is it also a consistent estimator ofη?

3. Let X1, . . . , Xn be IID beta(ν−1, 1), where 0 < ν <∞ is an unknown parameter.

(a) Derive the distribution of Yi = − lnXi, for 1 ≤ i ≤ n.

(b) What is the maximum likelihood estimator (MLE) of ν? Is the MLE of ν alsoa complete sufficient statistic for ν?

(c) What is the best 5% level critical region for testing H0 : ν ≤ 10 versus H1 :ν > 10? In what sense is it the best?

(d) How big a sample is needed so that the above test will attain a 10% probabilityof type II error at ν = 5.6?

1

(e) Develop a sequential probability ratio test for testing H0 : ν = 5 versus H2 :ν = 5.6 that will attain a 5% probability of type I error and a 10% probabilityof type II error.

4. A photocopy machine is fitted with a counter that records the number of copiesmade between successive breakdowns of the machine. After the repair person fixesthe machine she resets the counter to zero. Below are the data on number of copiesmade between successive breakdowns.

7638, 1037, 5982, 20292, 20132, 13110, 4438, 4075, 12517, 14869, 2571, 32891

The summary statistics are: n = 12, x = 11629.3, sx = 9360.1.

We are interested in finding a 95% confidence interval for μ, the average number ofcopies made before the machine breaks down. First, decide whether the data followan exponential distribution. If it does, find a parametric confidence interval; and ifit does not, find a non-parametric confidence interval for μ.

2