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Stat 643 Final Exam December 20, 2002 Prof. Vardeman 1. Suppose that \ µ NÐ)ß "Ñ where ) - @ œ d. Consider first a decision problem where T œ @ and PÐ)ß +Ñ œ a" • ) +b# Þ a) Find a generalized Bayes rule in this problem against Lebesgue measure on @. b) Argue that every finite nonrandomized decision rule $ ÐBÑ is Bayes versus ? a point mass prior concentrated at 0. c) Argue from first principles (you don't need a theorem from class for this) that $ ‡ ÐBÑ ´ ! is minimax in this decision problem. Consider now the squared error loss estimation of ") , i.e. a second decision problem with # @ œ d/Ö!×ß T œ @ and loss P‡ a)ß +b œ ˆ )" • +‰ Þ d) Argue that for - - Ð!ß "Ñ the estimator $- ÐBÑ œ B " € -B# is admissible for this second problem. •••••••••••••••••••••••••••••••••••••••••••••••••••••• Hint for part d): It is a fact that you need not prove that under the first loss function, P, the rule B $ÐBÑ œ # # " € B ˆ "€7 7 # ‰ is Bayes versus a Na0,7 # b prior. •••••••••••••••••••••••••••••••••••••••••••••••••••••• 2. Suppose that \ µ BernoulliÐ)Ñ where @ œ Ö "$ ß #$ ×Þ Consider a decision problem where T œ Ð!ß "Ñ (the unit interval) and " PÐ ß +Ñ œ +# $ # PÐ ß +Ñ œ " • + $ a) Argue that the set of nonrandomized decision rules is essentially complete in this problem. b) Consider a behavioral decision rule 9B where 9! is the Uniform Ð!ß "Ñ distribution 9" has pdf 0 Ð+Ñ œ #+MÒ! • + • "Ó Identify a nonrandomized rule that is better than it (and argue carefully that your candidate does the job). 1 c) Find the risk point ˆVÐ "$ ß $ ‰ß V Ð #$ ß $ ÑÑ corresponding to an arbitrary nonrandomized decision rule $ . d) Set up but do not try to solve an equation or equations needed to identify a least favorable prior distribution in this problem (a "1" in the notation below). •••••••••••••••••••••••••••••••••••••••••••••••••••••• Hint for part d): You may use the following without proof. For a prior distribution K, abbreviate 1 œ KˆÖ "$ ×‰ and " • 1 œ KˆÖ #$ ×‰. For $K Bayes versus K $K Ð!Ñ œ ž "•1 %1 if 1 ž " if 1 Ÿ " 5 " 5 and $K Ð"Ñ œ ž "•1 1 if 1 ž " if 1 Ÿ " # " # •••••••••••••••••••••••••••••••••••••••••••••••••••••• 3. For ) - @ œ Ö"ß #× let /) be NÐ)ß "Ñ measure and () be PoissonÐ)Ñ measure. You may take as given that for testing H! À ) œ " vs Ha À ) œ # based on \ µ /) , most powerful level ! tests reject for large \ and that the same is true for testing based on \ µ () . In fact, tables of normal and Poisson probabilities can be consulted to see that a best size ! œ Þ!) test in the first case rejects if \ ž #Þ%!& and in the second case rejects if \ &. Suppose that \ µ "# /) € "# () and that one wishes to test H! À ) œ " vs Ha À ) œ # based on \ . (Notice that with T) probability 1, one knows after observing \ whether it was the Normal distribution or the Poisson distribution that generated \ .) A possible test here is 9ÐBÑ œ MÒB Â m and B ž #Þ%!&Ó € MÒB - m and B &Ó a) Is this test of size ! œ Þ!)? Explain. b) Is this test most powerful of its size? Argue carefully one way or the other. 4. Here is a problem related to "scale counting." As a means of "automated" counting of a lot of discrete objects, I might weigh the lot and try to infer the number in the lot from the observed weight. Suppose that items have weights that are normally distributed with mean . and standard deviation ". In a calibration phase, I weigh a group of &! of these items (the number being known from a "hand count") and observe the weight \ . I subsequently weigh a large number, Q , of these items and observe the weight ] . I might model \ µ NÐ&!.ß &!Ñ independent of ] µ NÐQ .ß Q Ñ. Suppose further that I use a prior distribution for Ð.ß Q Ñ of independence, . µ NÐ!ß Ð"!!!Ñ# Ñ and Q • " µ PoissonÐ"!!Ñ. Describe in as much detail as possible (given the time constraints here) a simulation-based way of approximating the posterior mean of Q . 2