Mathematics Ph.D. Qualifying Examination Stat Probability, January 2018

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1 Mathematics Ph.D. Qualifying Examination Stat Probability, January 2018 NOTE: Answers all questions completely. Justify every step. Time allowed: 3 hours. 1. Let X 1,..., X n be a random sample from a normal distribution N(θ, 1). Answer the following questions. (1) Consider the hypotheses H 0 : θ = θ 0 against H 1 : θ > θ 0. 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 x n = , do you reject H 0 when θ 0 = 10 at the level α =.05 of significance? (2) Derive the likelihood ratio test at the level α (0, 1) of significance for testing H 0 : θ = θ 0 against H 1 : θ θ 0, where θ 0 is specified. (Give the rejection rule). (3) Is the test in (2) an unbiased test of H 0 against H 1? Justify your conclusion. (4) Is the test in (2) a uniformly most powerful test of H 0 against H 1? Justify your conclusion. 2. Let X 1,..., X n 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 H 0 : θ = θ, where θ is a fixed positive number, against the alternative composite hypothesis H 1 : 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) = log f(x; θ), the Fisher information I(θ) = E([S(θ; θ X)]2 ), and the Hessian function H(θ) = E( 2 log f(x; θ)) (provided that the latter two expected values are 2 θ 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 X 1,..., X n be i.i.d. with E(X 1 ) = µ and V ar(x 1 ) = σ 2 (0, ). Let X = (X X n )/n and Sn 2 = n i=1 (X i X) 2 /(n 1). Show T n = n( X µ) S n converges in distribution to the standard normal.

2 6. Suppose X 1,..., X n are independent and identically distributed with density f(x; θ) = θx θ 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

3 Mathematics Qualifying Examination January 2015 STAT Mathematical Statistics NOTE: Answer all questions completely and justify your derivations and steps. A calculator and statistical tables (normal, t and chi-square) are allowed. Time: 3 hours. 1. Suppose that the random vector = (X 1,...,X n ) X has the multi-normal distribution N n (µ,σ 2 I n ), with = (µ 1,...,µ n ) µ and I n is the n n identity matrix. Let X = 1 n 1 n, with 1 n = (1,1,...,1) X, be the usual average. a) Find the exact distribution of the quadratic form, Q = n(x) 2. b) Evaluate E(Q) when n = 10, σ 2 = 4 and µ i = 2, i. 2. Consider the SLR model Y i = β 0 +β 1 x i +ɛ i, with ɛ i N(0,σ 2 ), i.i.d., i = 1,2,...,n. Let η 0 E(Y x = x 0 ) = β 0 + β 1 x 0, be the mean response of Y evaluated at some fixed value x 0 of x. a) Derive the LSE ˆη 0 of η 0? b) Calculate the mean and variance of this estimator, ˆη 0? c) Obtain a (1 α) 100% confidence interval for η Let X 1,X 2,...,X n be a random sample from a continuous distribution whose p.d.f. is f(x λ,γ) = λe 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 = λ 0 is known, find the MLE for γ and obtain its pdf. Is it complete (prove or disprove)? c) Assuming that γ = γ 0 is known, find the MLE for λ and obtain its pdf. Is it complete (prove or disprove)? d) Find the (joint) MLE ˆθ n (ˆλ n, ˆγ n ) for θ = (λ,γ). e) Prove that ˆλ n and ˆγ n of part d) are independent r.v. s. f) Find the MLE, ˆψn, of the reliability ψ(λ,γ) = Pr(X t 0 λ,γ) at some known t 0 > γ. t 0 f(x λ,γ)dx, 1

4 4. Refer to problem 3 above. a) Construct an appropriate α-level likelihood ratio (LR) test of H 0 : λ λ 0 against H 1 : λ > λ 0 (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) = γ 2. Use results you obtained in problem 3 to derive an explicit expression for the UMVUE ˆξ n of ξ. 6. Suppose we have four identical coins and we would like to test H 0 : p 1/2 versus H 1 : 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 X i denote the number of Tails counted until the first Head occurs for the i th coin, i = 1,2,3,4. a) Construct, based on the data, X 1,...,X 4, a size α = 3/16 UMP test of H 0 versus H 1. b) What is the power of the above UMP test at p = 1/4 and p = 3/4? 2

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9 Mathematics Qualifying Examination August 2014 STAT Mathematical Statistics NOTE: Answer all questions completely and justify your derivations and steps. A calculator and statistical tables (normal, t and chi-square) are allowed. Time: 3 hours. 1. Let Y 1,Y 2,...,Y n be a random sample from a Bernoulli distribution with parameter θ, where θ is restricted to the interval Θ (0,3/4]. Find the MLE of ν = θ(1 θ). 2. Let X 1,X 2,...,X n be i.i.d. observations from the gamma distribution f(x λ) = Xα 1 e x/λ Γ(α)λ α, 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 t 0 > 0 given γ(λ) = t 0 f(t λ)dt. c) Find the Cramer-Rao lower bound for the variance of an unbiased estimator of γ(λ) based on X 1,...,X n or a suitable transformation thereof. d) Is the MLE of γ(λ) consistent and asymptotically efficient? Support your assertions. 3. Let X 1,X 2,...,X n be a random sample from N(0,σ 2 ). a) Find the UMVUE for σ 2. b) Show that your answer to part a above is statistically independent of X (1) X (2), the ratio of the first two order statistics of the given sample. 4. Suppose that the independent random variables Y 1,Y 2,...,Y n satisfy Y i = βx i + ɛ i, for, i = 1,2,...,n, where x 1,...,x n are some known constants, β is an unknown regression parameter, and ɛ i N(0,σ 2 ), are iid and σ 2 is a known constant. a) Construct the Likelihood Ratio Test (LRT) of H 0 : β = 0, versus H 1 : β 0. b) Suppose that n = 100 and x i = 10 and σ 2 = 5. Find the exact rejection region 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 H 0 : p = 0.5, versus H 1 : p > 0.5, where p is the unknown probability that any one of these coins comes 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 X i denote the number of tails that occur before the first head occurs when the i th coin is flipped, i = 1,2,3,4. Use these data, X 1,X 2,X 3,X 4 to construct a Uniformly Most Powerful (UMP) test of size α = 3/16 of H 0 versus H 1. 1

10 6. Consider the problem of testing, based on a sample of size 1, of the hypothesis H 0 :X N(0,1) against the alternative hypothesis H 1 :X εn(0,1)+(1 ε)c(0,1) for some unknown ε > 0, where C(0,1) stands for the standard Cauchy distribution with 1 density π(1+x 2 ), < x <. a) Are H 0 and H 1 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 distribution with variance 1 equals zero (which is to reject H 0 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 M p {g : E p (g(x)) = 0 and V ar p (g(x)) = 1 }. Find all members of the class of functions M p. (Hint: There are two such members in M p.) d) For X and Y as above, define the maximal correlation over M p as ρ M Calculate ρ M and compare it to ρ XY. sup Cor(g(X),u(Y )). g,u M p 2

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15 Mathematics Ph.D. Qualifying Examination January 2013 STAT Mathematical Statistics NOTE: Answer all four questions completely. Justify every step. A calculator and some statistical tables (normal, t and chi-square) are allowed. Time allowed: 3 hours. 1. Suppose that X 1,...,X n is a random sample from the probability density function f(x θ) = 2 x /θ, 0 <x<, θ e x2 where θ>0 is an unknown parameter. (a) Find the method of moments estimator (MME) of θ and the maximum likelihood 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% confidence interval for θ. 2. Let Y 1 < Y 2 <... < Y n be the order statistics of a random sample from a lognormal(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 ˆψn is either the MVUE or the MLE of ψ, show that, n r ( ψ n ψ) 0in probability as n, for any r [0, 0.5). (d) Derive an unbiased estimator η n of η =Φ( θ), where Φ( ) isthecumulative distribution function of the standard normal variable. Starting from this η n, how will you derive the MVUE of η? 3. Let X 1,...,X n 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? IstheMLEofν also a complete sufficient statistic for ν? (b) What is the best 5% level critical region for testing H 0 : ν 5versusH 1 : 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% probability of type II error at ν =6? 1

16 (d) Develop a sequential probability ratio test for testing H 0 : ν =5versusH 2 : ν = 6 that will attain a 5% probability of type I error and a 10% probability of type II error. 4. (a) The table below gives the number of days spent in the ICU by all patients admitted during the week of December 23 29, This information is gathered after all such patients have been discharged from the ICU. Table 1: Duration of Stay in ICU #Days total #Patients Construct a 95% confidence interval for μ, the average number of days spent in the ICU by each patient. (b) Now suppose that we consider a different survey design. The surveyor visits the ICU at noon of each Sunday in December 2012 and makes a list of all patients in the 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 total #Patients Based on the data in Table 2, construct a 95% confidence interval for μ. 2

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27 Mathematics Ph.D. Qualifying Examination August 2012 STAT Mathematical Statistics NOTE: Answer all four questions completely. Justify every step. A calculator and some statistical tables (normal, t and chi-square) are allowed. Time allowed: 3 hours. 1. Suppose that X 1,...,X n is a random sample from the probability density function f(x θ) = rxr 1 e xr /θ, 0 <x<, θ 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% confidence interval for θ. 2. Let X 1,...,X n be a random sample from a Poisson(λ) distribution, where λ>0is the mean parameter. (a) Find the uniformly minimum variance unbiased estimator (UMVUE) ψ n and the maximum likelihood estimator (MLE) ˆψ n of ψ = e λ. (b) Compare the UMVUE and the MLE of ψ. Are the two similar or quite different? Explain. (c) Show that, for any r [0, 0.5), the UMVUE satisfies n r ( ψ n ψ) 0in probability as n. (d) Derive the UMVUE η n of η = P {X i 1}. Is it also a consistent estimator of η? 3. Let X 1,...,X n be IID beta(ν 1, 1), where 0 <ν< is an unknown parameter. (a) Derive the distribution of Y i = ln X i,for1 i n. (b) What is the maximum likelihood estimator (MLE) of ν? IstheMLEofν also a complete sufficient statistic for ν? (c) What is the best 5% level critical region for testing H 0 : ν 10 versus H 1 : 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% probability of type II error at ν =5.6? 1

28 (e) Develop a sequential probability ratio test for testing H 0 : ν =5versusH 2 : ν =5.6 that will attain a 5% probability of type I error and a 10% probability of type II error. 4. A photocopy machine is fitted with a counter that records the number of copies made between successive breakdowns of the machine. After the repair person fixes the machine she resets the counter to zero. Below are the data on number of copies made between successive breakdowns. 7638, 1037, 5982, 20292, 20132, 13110, 4438, 4075, 12517, 14869, 2571, The summary statistics are: n =12, x = ,s x = We are interested in finding a 95% confidence interval for μ, the average number of copies made before the machine breaks down. First, decide whether the data follow an exponential distribution. If it does, find a parametric confidence interval; and if it does not, find a non-parametric confidence interval for μ. 2

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