PhD Qualifying Examination Department of Statistics, University of Florida

Transcription

1 PhD Qualifying xamination Department of Statistics, University of Florida January 24, 2003, 8:00 am - 12:00 noon Instructions: 1 You have exactly four hours to answer questions in this examination 2 There are 8 problems of which you must answer 6 3 Only your first 6 problems will be graded 4 Write only on one side of the paper, and start each question on a new page 5 Write your number on every page 6 Do not write your name anywhere on your exam 7 You must show your work to receive credit 8 While the eight questions are equally weighted, within a given question, the parts may have different weights The following abbreviations are used throughout: GLM = generalized linear model mgf = moment generating function UMP = uniformly most powerful 1

2 1 Let {X n, n 1} be a sequence of random variables and let S n = n =1 X, n 1 Prove that if n=1 X n <, then there exists a random variable S with S n S almost certainly and S n L 1 S 2 Let {X n, n 1} be a sequence of independent and identically distributed random variables with X 1 q < for some q (0, ) Prove that for all p (1, ), n =1 X pq n p 0 almost certainly 3 (a) Suppose that L Poisson(φ) and that Y L χ 2 q+2l ; that is, conditional on L, Y has a χ2 distribution with q+2l degrees of freedom Write down the marginal density function of Y You should recognize this as the non-central χ 2 distribution with q degrees of freedom and non-centrality parameter φ (b) Find the mean of Y (c) Let X 1,, X p be independent random variables such that X i N(θ i, 1) for i = 1,, p Assume that p > 2 Put X = (X 1,, X p ) T, θ = (θ 1,, θ p ) T and λ = θ 2 /2 Show that ( ) 1 X 2 = [g(k)] (1) where K Poisson(λ) In other words, identify the function g (In order to answer this question, you need to know the distribution of X 2 However, you are not required to derive this distribution) (d) The equation (1) can clearly be rewritten as { } 1 R p x 2 exp 1 p (x (2π) p/2 i θ i ) 2 dx = 2 Assuming that θ show that i=1 k=0 exp{ λ} λk g(k) k! can be passed through the integral and through the sum, differentiate both sides to ( ) X X 2 = θ ( ) λ K p 2 + 2K (e) Consider the James-Stein estimator of θ given by ( δ(x) = 1 p 2 ) X 2 X Use the above results to show that the mean squared error of δ can be written as ( ) δ(x) θ 2 = p (p 2) 2 1 p 2 + 2K (f) In the context of estimating θ under squared error loss, what have we shown? 2

3 4 (a) Suppose Z θ Geometric(θ); that is, P (Z = z θ) = θ(1 θ) z for z Z + = {0, 1, 2, } and θ (0, 1) Note that (Z θ) = 1 θ θ marginal mass function of Z assuming that θ Beta(α, β) d and Var(Z θ) = 1 θ Find the θ 2 (b) The function ψ(x) = dx log Γ(x) (defined for positive x) is called the digamma function The digamma function has the following integral representation ψ(x) = γ t x 1 1 t where γ is uler s constant Use this representation to show that ψ(x) is an increasing function (Hint: You don t need any derivatives) (c) Now use the fact that ψ is increasing to show that for fixed 0 < a < b, the function g(t) = is decreasing in t (Hint: Use a log and a derivative) Γ(t + a) Γ(t + b) (d) Suppose we have a single observation from the mass function dt, P α (Z = z) = α2 Γ(α) z! Γ(z + α + 2) for z Z + Construct a UMP size 010 test of H 0 : α 3 versus H A : α > 3 (Hint: You are not being forced to use the Neyman-Pearson Lemma here) 5 Let Y = (Y 1,, Y k ) be a multinomial vector of counts based on m trials and probability vector π = (π 1,, π k ) (a) Show that the oint mgf of Y is HINT: Use the identity, k M Y (t) = π e t k =1 α m =1 = y S where S = {y = (y 1,, y k ) y 0, y = m} (b) Derive the mean vector and covariance matrix of Y m! y! m α y, (c) Let Y 1,, Y n be independent multinomial vectors each with k categories Suppose that Y i is based on m i trials and probability vector π i = (π i1,, π ik ), i = 1,, n Suppose further that the π i s satisfy the model, exp(x i π i = β) k r=1 exp(x ir β), where x i is a vector of known covariates associated with the (i, )th count Write down the loglikelihood function for the parameter β Show that there exists a Poisson loglinear GLM for which likelihood inference concerning β is identical to that based on this multinomial model 3

4 6 Suppose that Y has a binomial distribution with m trials and probability π (a) xpress the binomial likelihood function in exponential form in terms of the canonical parameter θ = logit(π) (b) Derive the deviance measure of fit D(y, µ) for the binomial model, where µ = mπ (c) Show that the deviance can be approximated by the Pearson χ 2 statistic, X 2, if m is large, where X 2 = m(p π)2 π(1 π), and p = Y/m (d) Argue that, for c > 0 Hence show that {log(y + c)} = log(mπ) + c mπ 1 π 2mπ + O(m 3/2 ) { ( )} Y + c log = θ + (1 2π)(c 1 2 ) + O(m 3/2 ) m Y + c mπ(1 π) (e) Comment briefly on the relevance of the result in (d) 7 Let x Ax be a quadratic form in x which is distributed as N(µ, Σ) (a) Give a complete expression for φ(t), the mgf of x Ax (b) Show that φ(t) exists if t < c for some constant c (specify what c is) (c) Make use of (a) to show that if AΣ is idempotent of rank r, then x Ax is distributed as χ 2 r (λ) Please specify what the non-centrality parameter is 4

5 8 Consider the model y = Xβ + ɛ, where X is n p of rank r(< p), ɛ N(0, σ 2 I n ) Let M be an s-dimensional subspace of the row space of X(s r) and let C be a matrix of order s p and rank s whose rows form a basis for M It is known that Scheffé s simultaneous (1 α)100% confidence intervals on all estimable linear functions of the form a β, where a M, are given by a ˆβ ± { s[a (X X) a]ms F α,s,n r } 1/2, (2) where MS is the error mean square and ˆβ = (X X) X y (You do not have to prove (2)) (a) The F -test concerning the hypothesis H 0 : C β = 0 is significant at the α-level if and only if there exists a 0 M such that a 0 ˆβ > {s[a 0(X X) a 0 ]MS F α,s,n r } 1/2 (3) (b) Write a 0 in inequality (3) as a 0 = b 0C, where b 0 is some vector in R s, the s-dimensional uclidean space Show that inequality (3) is equivalent to (c) Show that inequality (4) can be written as where G 1 = C ˆβ ˆβ C, G2 = C (X X) C (d) Show that b C ˆβ b C (X X) Cb 1/2 > (sms F α,s,n r ) 1/2 (4) s b G 1 b b G s 2 b > s MS F α,s,n r, b G 1 b b G s 2 b = e max(g 1 2 G 1), where e max (G 1 2 G 1) is the largest eigenvalue of G 1 2 G 1 (e) Show that b G 1 b b G 2 b attains its remum if b is an eigenvector of G 1 2 G 1 corresponding to e max (G 1 2 G 1) 5