NAME 18 466 nal 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 1 1 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 What is the power of each test in part d 2 2 a De ne sequential probability ratio tests SPRTs b State the main optimality theorem about SPRTs 3 3 a De ne exponential families b De ne the order of an exponential family c For n i i d observations from an exponential family as de ned in part a give a su cient statistic whose dimension equals the order and so doesn t depend on n d For the family of gamma distributions with densities f x x 1 e x for x 0 and 0 for x 0 where 0 and 0 show that the family is exponential and nd its order e What is the statistic as in part c for n i i d observations 4 4 Let a parameter range over an interval a b Let T be an estimator of a function g with a bias b so that E T g b for a b Here T g and b are real valued Suppose su cient conditions for the information inequality hold for T as an unbiased estimator of g b Give a lower bound for the mean squared error of T as an estimator of g E T g 2 5 5 Let x have a N I distribution in R3 where I is the 3 3 identity matrix and the unknown can be any vector in R3 a What is E x 2 b What is a function J x such that E J x 2 E x 2 for all in R3 6 n 2 6 Let X1 Xn be i i d N 2 Let V j 1 Xj X a For what constants c1 n depending on n is c1 n V an unbiased estimator of 2 b For what constants c2 n depending on n is the mean square error E 2 c1 n V 2 2 minimized for all 0 Hints V 2 has a 2n 1 distribution A 2d distribution is as in problem 3 d with d 2 and 1 2 If Y has a distribution then EY and Y has variance 2 7 7 Let be a parameter space and X B a sample space Given a function h x on X a De ne M estimator of type based on h b De ne what is a sequence Tn Tn X1 Xn of approximate M estimators based on h c De ne what it means for h to be adjusted d De ne what it means for h to be adjustable e If observations have a distribution P de ne what it means for a 0 to be pseudotrue f What is the relationship between a sequence Tn of approximate M estimators and a pseudo true 0 under su cient regularity conditions 8 8 a De ne 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 x log f x for the likelihood function f of a family P and let the distribution P of the data be P 1 for some 1 What is an adjustment function for h in this case d If the conditions in c hold is the true 1 always sometimes or never equal to a pseudo true value 0 Explain 9 9 a De ne the nite sample breakdown point of a statistic Tn Tn X1 Xn having values in a parameter space 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 De ne 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 10 10 Suppose given a family P where is an open subset of a Euclidean space Rd with a likelihood function f x 0 for all and x a De ne the Fisher information matrix I of the family b If f x is C 2 as a function of and other suitable conditions hold give an alternate form of I c Let Tn Tn X1 Xn be a sequence of estimators of When are Tn said to be e cient d What class of estimators were shown in the course to be e cient under some conditions 11 11 In the previous problem suppose d 1 and we want to estimate a function g a Under some regularity conditions what is an asymptotic lower bound for the meansquare error E n Tn g 2 b Does the asymptotic lower bound hold for all or if not what can be said about those for which it doesn t c Give an example where the regularity conditions needed for a fail and g can be estimated with a mean square error that goes to 0 faster as n 12 12 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 13 13 A beta a b distribution has a density xa 1 1 x b 1 B a b for 0 x 1 and 0 1 elsewhere where a 0 b 0 and B a b is the beta function B a b 0 ta 1 1 t b 1 dt 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 0 and b 0 a After n observations what is the likelihood function f p X1 Xn b What is the posterior distribution of p c What does it mean for posteriors to be consistent d Are …
View Full Document