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MIT 18 466 - Final Exam

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NAME: 18.466 final exam, Wednesday, May 21, 2003, 9 A.M.-noon Closed book exam. No books or notes may be consulted during this exam. There are 13 questions on the exam. Answer any TEN of the 13 for full credit. Please indicate which three you omit. Explanations should be given only where requested, or as time permits. 11. Let X = {0, 1, 2},P(0) = 0.8,P(1) = 0.05,P(2) = 0.15,Q(0) = 0.008,Q(1) = 0.002,Q(2) = 0.99. (a) What is the most powerful non-randomized test of P vs. Q with size ≤ 0.05? (b) What is the power of the test in (a)? (c) Find the likelihood ratio RQ/P (x)at each x. (d) What are the admissible non-randomized tests of P vs. Q? (e)Whatisthe power ofeachtestin part(d)? 22. (a) Define sequential probability ratio tests (SPRTs). (b) State the main optimality theorem about SPRTs. 33. (a) Define exponential families. (b) Define the order of an exponential family. (c) For n i.i.d. observations from an exponential family as defined in part (a), give a sufficient statistic whose dimension equals the order and so doesn’t depend on n. (d) For the family of gamma distributions with densities α−1f((α, λ),x)= λα x e −λx /Γ(α) for x> 0and 0for x ≤ 0, where α> 0and λ> 0, show that the family is exponential and find its order. (e)Whatisthe statistic as in part(c) for n i.i.d. observations? 44. Let a parameter θ range over an interval a<θ<b.Let T be an estimator of a function g(θ)with a bias b(θ), so that Eθ T = g(θ)+ b(θ)for a<θ < b.Here T , g,and b are real-valued. Suppose sufficient conditions for the information inequality hold for T as an unbiased estimator of (g + b)(θ). Give a lower bound for the mean-squared error of T as an estimator of g(θ), Eθ ((T − g(θ))2). 55. Let x have a N(µ, I) distribution in R3 where I is the 3 × 3identitymatrixand the unknown µ can be any vector in R3 . (a)Whatis Eµ|x − µ|2? (b) What is a function J(x) such that Eµ|J(x) − µ|2 <Eµ|x − µ|2 for all µ in R3? 6�n6. Let X1,... ,Xn be i.i.d. N(µ, σ2). Let V := j=1(Xj − X)2 . (a) For what constants c1(n) depending on n is c1(n)V an unbiased estimator of σ2? (b) For what constants c2(n) depending on n is the mean-square error Eσ2[(c1(n)V − σ2)2] minimized for all σ> 0? Hints: V/σ2 has a χ2 distribution. A χ2 distribution is n−1 d Γ(α, λ) as in problem 3(d) with α = d/2and λ =1/2. If Y has a Γ(α, λ) distribution then EY = α/λ and Y has variance α/λ2 . 77. Let Θ be a parameter space and (X, B) a sample space. Given a function h(θ, x)on Θ × X, (a) Define M-estimator (of ρ type) based on h. (b) Define what is a sequence Tn = Tn (X1,... ,Xn) of approximate M-estimators based on h. (c) Define what it means for h to be adjusted. (d) Define what it means for h to be adjustable. (e) If observations have a distribution P , define what it means for a θ0 ∈ Θ to be pseudo-true. (f) What is the relationship between a sequence Tn of approximate M-estimators and a pseudo-true θ0 under sufficient regularity conditions? 88. (a) Define the Kullback-Leibler information I(P, Q)for two laws P and Q on a sample space. (b) Under what conditions on laws P and Q is: I(P, Q) < 0? I(P, Q)= 0? I(P, Q) > 0? (c) Let h(θ, x)= − log f(θ, x) for the likelihood function f of a family {Pθ ,θ ∈ Θ},and let the distribution P of the data be Pθ1 for some θ1 ∈ Θ. What is an adjustment function for h in this case? (d) If the conditions in (c) hold, is the true θ1 always, sometimes, or never equal to a pseudo-true value θ0? Explain. 99. (a) Define the finite-sample breakdown point of a statistic Tn = Tn (X1,... ,Xn)having values in a parameter space Θ at a sample (X1,... ,Xn). (b) For real observations (X1,... ,Xn) and their order statistics X(1) ≤ ··· ≤ X(n),what is the breakdown point of X(j)? (c) Define equivariance for location of a real-valued statistic Tn = Tn((X1,... ,Xn)) for real (X1,... ,Xn). (d) What can be said about the breakdown points of statistics equivariant for location? (e) What is an example of a sequence of statistics Tn , each equivariant for location, having largest possible breakdown point for each n? 1010. Suppose given a family {Pθ ,θ ∈ Θ} where Θ is an open subset of a Euclidean space Rd, with a likelihood function f(θ, x) > 0 for all θ and x. (a) Define the Fisher information matrix I(θ) of the family. (b) If f(θ, x)is C2 as a function of θ and other suitable conditions hold, give an alternate form of I(θ). (c) Let Tn = Tn((X1,... ,Xn) ) be a sequence of estimators of θ.When are Tn said to be efficient? (d) What class of estimators were shown in the course to be efficient under some conditions? 1111. In the previous problem, suppose d = 1 and we want to estimate a function g(θ). (a) Under some regularity conditions, what is an asymptotic lower bound for the mean-square error Eθ [n(Tn − g(θ))2]? (b) Does the asymptotic lower bound hold for all θ or if not, what can be said about those θ for which it doesn’t? (c) Give an example where the regularity conditions needed for (a) fail and g(θ)can be estimated with a mean-square error that goes to 0 faster as n →∞. 1212. (a) For what kinds of hypotheses and alternatives does the Wilks test provide a test? (b) What is the Wilks test statistic? (c) What is the asymptotic distribution of the statistic, under some regularity conditions, and if which hypothesis is true? 1313. A beta(a, b) distribution has a density xa−1(1 − x)b−1/B(a, b)for 0 <x< 1and 0 elsewhere, where a> 0, b> 0, and B(a, b) is the beta function B(a, b)= � 1 ta−1(1−t)b−1dt.0 The beta(a, b) distribution has mean a/(a + b)and variance ab/[(a + b)2(a + b +1)]. Let X1,X2,... , be i.i.d. with a Bernoulli distribution, P (Xj =1)= p =1 − P (Xj =0) where 0 ≤ p ≤ 1. Let Sn = X1 + ···+ Xn.Let p have a beta(a, b) prior for some a> 0and b> 0. (a) After n observations, what is the likelihood function f(p, ((X1,... ,Xn) ))? (b) What is the posterior distribution of p? (c)Whatdoesitmean for posteriors to be consistent? (d) Are the posteriors consistent in this case: for any p,0 ≤ p ≤ 1, or only if 0 <p< 1, possibly depending on a and b?

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