Stat 643 Problems, Fall 2000

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1 Stat 643 Problems, Fall 2000 Assignment 1 1. Let ( HT, ) be a measurable space. Suppose that., T and Uare 5-finite positive measures on this space ith T Uand U...T.T.U a) Sho that T. and œ a.s......u.....u.u Š.. b) Sho that if. U, then œ a.s (Cressie) Let H œò -- ß Ó for some -, T be the set of Borel sets and T be Lebesgue measure divided by 2 -. For a subset of H, define the symmetric set for E as EœÖ == l E and let V œöe T Eœ E Þ a) Sho that V is a sub 5-algebra of T. b) Let be an integrable random variable. Find EÐlVÑÞ 3. Let H œòßó, T be the set of Borel sets and Tœ?. for? a distribution placing a unit point mass at the point Ð ßÑ and. 2-dimensional Lebesgue measure. Consider the variable Ð= Ñœ = and the sub 5-algebra of T generated by, V. a) For E T, find E ÐMElVÑœTÐElVÑ. b) For ]Ð= Ñœ =, find E Ð]lVÑ. 4. Suppose that œð ßßÞÞÞßÑ has independent components, here each 3 is generated as follos. For independent random variables [ 3 µ normal Ð. ßÑ and ^ 3 µ Poisson Ð. Ñ, 3 œ[ 3 ith probability : and 3 œ^ 3 ith probability :. Suppose that. Òß_Ñ. Use the factorization theorem and find lo dimensional sufficient statistics in the cases that: a) : is knon to be, and b) : ÒßÓ is unknon. (In the first case the parameter space œö Òß_Ñ, hile in the second it is ÒßÓ Òß_Ñ.) 5. (Ferguson) Consider the probability distributions on Ðe ßU Ñ defined as follos. For ) œð) œ e ÐßÑ and µt, suppose TÒ œbó œ ) œ ) B ) Ð :Ñ: for Bœ )) ß ß) ßÞÞÞ otherise. Let ßßÞÞÞß be iid according to TÞ ) a) Argue that the family ÖT ) is not dominated by a 5-finte measure, so that the factorization theorem can not be applied to identify a sufficient statistic here. b) Argue from first principles that the statistic XÐÑ œð min ßÑis sufficient for the parameter )

2 c) Argue that the factorization theorem can be applied if ) is knon (the first factor of the parameter is replaced by a single point) and identify a sufficient statistic for this case. d) Argue that if : is knon, min 3 is a sufficient statistic. 6. Suppose that is exponential ith mean - (i.e. has density 0ÐBÑ - œ -exp Ð -BÑMÒB Ó ith respect to Lebesgue measure on e, but that one only observes œmò Ó. (There is interval censoring belo Bœ.) a) Consider the measure. on k œö Ðß_Ñ consisting of a point mass of 1 at 0 plus Lebesgue measure on (1,_). Give a formula for the R-N derivative of T - rt. on k. b) Suppose that ßßÞÞÞß are iid ith the marginal distribution T-. Find a 2-dimensional sufficient statistic for this problem and argue that it is indeed sufficient. c) Argue carefully that your statistic from b) is minimal sufficient. 7. As a complement to Problem 6, consider this random censoring model. Suppose that is as in Problem 6. Suppose further that independent of, ^ is exponential ith mean. We ill suppose that ith [œ minaß^b one observes œ amò[ œóß[ b (one sees either or a random censoring time less than ). The family of distributions of is absolutely continuous ith respect to the product of counting measure and Lebesgue measure on k œöß e (call this dominating measure.). a) Find an R-N derivative of T - rt. on k. b) Supose that ßßÞÞÞß are iid ith the marginal distribution T-. Find a minimal sufficient statistic here. (Actually carefully proving minimalsufficiency looks like it ould be hard. Just make a good guess and say hat needs to be done to finish the proof.). Schervish Exercise 2.. And sho that ^ is ancillary. 9. Schervish Exercise (I think you need to assume that ÖB spans e.) 10. Schervish Exercise Schervish Exercise What is a first order ancillary statistic here? 12. Schervish Exercise

3 Assignment Schervish Exercise 2.34 (consider iid observations). 14. Schervish Exercise Schervish Exercise Schervish Exercise Schervish Exercise Prove the folloing version of Theorem 24 (that Vardeman gave only a discrete case argument for) À Suppose that œðxßwñ and the family c of distributions of on k œ g f is dominated by the measure. œ / (a product of 5-finite measures on g and f.t respectively). With Ð>ß=Ñ œ0ð>ß=ñ, let. ). ) 1Ð>Ñ œ 0Ð>ß=Ñ. Ð=Ñ 0Ð=l>Ñ œ 0Ð>ß=Ñ ) ) ( ) and ) 1Ð>Ñ f Suppose that c is FI regular at ) and that 1Ð>Ñ œ ' 0Ð>ß=Ñ. Ð=Ñ a>. Then ) f ) M Ð) Ñ MX Ð) Ñ ) 19. Let T and T be to distributions on k and 0 and 0 be densities of these ith respect to some dominating 5 -finite measure.. Consider the parametric family of distributions ith parameter ) ÒßÓ and densities rt. of the form 0ÐBÑ ) œð ) Ñ0 ÐBÑ ) 0ÐBÑ ß and suppose that has density 0 ). a) If KÐÖ Ñ œkðö Ñ œ, find the posterior distribution of ) l. b) Suppose no that K is the uniform distribution on ÒßÓ. Find the posterior distribution of )l. i) What is the mean of this posterior distribution (one Bayesian ay of inventing a point estimator for :)? ii) Consider the partition œòßþ&ó œðþ&ßþó. One Bayesian ay of inventing a test for H is to decide in favor if the posterior probability assigned is at least.5. Describe as explicitly as you can the subset of k here this is the case. 3

4 20. Consider the situation of problem 6 and maximum likelihood estimation of -. a) Sho that ith Qœ Ò 3's equal to Ó, in the event that Qœthere is no MLE of -, but that in all other cases there is a maximizer of the likelihood. Then argue that for any -, ith T probability tending to 1, the MLE of -, say - ^ -, exists. b) Give a simple estimator of - based on Q alone. Prove that this estimator is consistent for -. Then rite don an explicit one-step Neton modification of your estimator from a). c) Discuss hat numerical methods you could use to find the MLE from a) in the event that it exists. d) Give to forms of large sample confidence intervals for - based on the MLE - ^ and to different approximations to MÐ-ÑÞ 21. Consider the situation of Problems 6 and 20. Belo are some data artificially generated from an exponential distribution..24, 3.20,.14, 1.6,.5, 1.15,.32,.66, 1.60,.34,.61,.09, 1.1, 1.29,.23,.5,.11, 3.2, 2.53,. a) Plot the loglikelihood function for the uncensored data (the B values given above). Give approximate 90% to-sided confidence intervals for - based on the asymptotic ; distribution for the LRT statistic for testing H À - œ - vs H À - Á - and based on the asymptotic normal distribution of the MLE. No consider the censored data problem here any value less than 1 is reported as 0. Modify the above data accordingly and do the folloing. b) Plot the loglikelihood for the censored data (the derived B) values. Ho does this function of - compare to the one from part a)? It might be informative to plot these on the same set of axes. c) It turns out (you might derive this fact) that expð -Ñ MÐ -Ñœ Ð Ñ Þ Œ ˆ - exp - - expð -Ñ Give to different approximate 90% confidence intervals for - based on the asymptotic distribution of the MLE here. Then give an approximate 90% interval based on inverting the LRTs of H À - œ - vs H À - Á- Þ 22. Suppose that ßß $ and % are independent binomial random variables, 3 µ binomial Ðß:Ñ 3. Consider the problem of testing H À: œ: and : $ œ: % against the alternative that H does not hold. a) Find the form of the LRT of these hypotheses and sho that the log of the LRT statistic is the sum of the logs of independent LRT statistics for H À: œ: and H À: $ œ: % (a fact that might be useful in directly shoing the ; limit of the LRT statistic under the null hypothesis). b) Find the form of the Wald tests and sho directly that the test statistic is asymptotically ; under the null hypothesis. 4

5 c) Find the form of the ; (score) tests of these hypotheses and again sho directly that the test statistic is asymptotically ; under the null hypothesis. 23. Suppose that ßßÞÞÞ are iid, each taking values in k œößß ith R-N derivative of rt to counting measure on k T ) 0ÐBÑ œ ) expðb) Ñ exp) expð ) Ñ 3 3œ a) Find an estimator of ) based on œ MÒ œó that is È consistent (i.e. for hich ÈÐs ) ) Ñ converges in distribution). b) Find in more or less explicit form a one step Neton modification of your estimator from a). c) Prove directly that your estimator from b) is asymptotically normal ith variance ÎM Ð) Ñ. (With ) ^ the estimator from a) and ) the estimator from b), ) ) œs PÐs Ñ ), PÐ s) Ñ and rite PÐs ÑœPÐÑ Ðs ) ) ) ) ÑP ÐÑ ) Ðs) ) ÑP Ð) Ñ for some ) s beteen ) and ).) d) Sho that provided B ÐßÑ the loglikelihood has a maximizer B $B 'B )s œ log È Þ Ð BÑ Prove that an estimator defined to be )s hen B ÐßÑ ill be asymptotically normal ith variance ÎM Ð) Ñ. e) Sho that the observed information and expected information approximations lead to the same large sample confidence intervals for ). What do these look like based on, say,? )s By the ay, a version of nearly everything in this problem orks in any one parameter exponential family. 24. Prove the discrete version of Theorem

6 Assignment (Optional) Consider the problem of Bayesian inference for the binomial parameter :. In particular, for sake of convenience, consider the Uniform ÐßÑ (Beta Ð ß Ñ for œ œ) prior distribution. (a) It is possible to argue from reasonably elementary principles that in this binomial context, œðßñ, the Beta posteriors have a consistency property. That is, simple arguments can be used to sho that for any fixed any % 0, for µ binomial Ðß:Ñ, the random variable : % Ð Ñ ] œ ( : Ð :Ñ.: FÐ ß Ð ÑÑ : % (hich is the posterior probability assigned to the interval Ð: % ß: % Ñ) converges in : probability to 1 as p_. This problem is meant to lead you through this argument. Let % 0 and $ 0. B B i) Argue that there exists 7 such that if 7, ¹ ¹ ab œßßþþþßþ Ð BÑÐ BÑ ii) Note that the posterior variance is Ð Ñ Ð ÑÞ Argue there is an 7 such that if 7 the probability that the posterior assigns to B % B % Š 3ß 3 is at least $ ab œßßþþþß. iii) Argue that there is an 7 such that if 7 the : probability that : % is at least $. $ Then note that if maxð7ß7ß7ñi) and ii) together imply that the posterior probability assigned to ˆ B % B % $ ß $ is at least $ for any realization B. Then provided B : % $ the posterior probability assigned to Ð: % ß: % Ñ is also at least $ Þ But iii) says this happens ith : probability at least $. That is, for large, ith : probability at least $, ] $ Þ Since $ is arbitrary, (and ] Ÿ ) e have the convergence of ] to 1 in probability. : (b) Corollary 45 suggests that in an iid model, posterior densities for large tend to look normal (ith means and variances related to the likelihood material). The posteriors in this binomial problem are Beta Ð Bß Ð BÑÑ (and e can think of µ binomial Ðß:Ñ as derived as the sum of iid Bernoulli Ð: Ñ variables). So e ought to expect Beta distributions for large parameter values to look roughly normal. To illustrate this (and thus essentially hat Corollary 45 says in this case), do the folloing. For 3 œþ$ (for example... any other value ould do as ell), consider the Beta Ð 3 ß Ð 3ÑÑ (posterior) distributions for œßß% and. For : µ Beta Ð 3 ß Ð 3ÑÑ plot the probability densities for the variables % 3 : and 6

7 Ê a b Ð Ñ : on a single set of axes along ith the standard normal density. Note that if [ has pdf 0Ð Ñ, then +[, has pdf 1Ð Ñœ 0ˆ, + +. (Your plots are translated and rescaled posterior densities of : based onpossible observed values B œþ$.) If this is any help in doing this plotting, I tried to calculate values of the Beta function using MathCad and got the folloing: $ & ÐFÐ%ß)ÑÑ œþ$ ßÐFÐ(ß&ÑÑ œ)þ% ß ( ÐFÐ$ß*ÑÑ œþ* and ÐFÐ$ß(ÑÑ œþ*'(. 26. Problem 12, parts a)-c) Schervish, page Consider the folloing model. Given parameters - ßÞÞÞß -R variables ßÞÞÞßÞ R are independent Poisson variables, 3 µ PoissonÐ-3ÑÞ Qis a parameter taking values in ÖßßÞÞÞßR and if 3ŸQß - 3 œ., hile if 3 Qß - 3 œ. Þ ( Q is the number of Poisson means that are the same as that of the first observation.) With parameter vector ) œðqß. ß. Ñ belonging œößßþþþßr Ðß_Ñ Ðß_Ñ e ish to make inference about Q based on ßÞÞÞßÞ in a Bayesian frameork. R R 3œ As matters of notational convenience, let W œ and XœW. a) If, for example, a prior distribution K is constructed by taking Q uniform on ÖßßÞÞÞßR independent of. exponential ith mean 1, independent of. exponential ith mean 1 ß it is possible to explicitly find the (marginal) posterior of Q given that œbßþþþß R œbr. Don't actually bother to finish the somehat messy calculations needed to do this, but sho that this is possible (indicate clearly hy appropriate integrals can be evaluated explicitly). b) Suppose no that K is constructed by taking Q uniform on ÖßßÞÞÞßR independent of Ð. ß. Ñ ith joint density 1Ð ß Ñrt Lebesgue measure on (0,_) (0,_). Describe in as much detail as possible a SSS based method for approximating the posterior of Q, given œbßþþþß R œbr. (Give the necessary conditionals up to multiplicative constants, say ho you're going to use them and hat you'll do ith any vectors you produce by simulation.) 2. Consider the simple to-dimensional discrete distribution for ) œð) ß) Ñ given in the table belo. 7

8 ) ) a) This is obviously a very simple setup here anything one might ish to kno or say about the distribution is easily derivable from the table. Hoever, for sake of example, suppose that one ished to use SSS to approximate UœTÒ) ŸÓ œþ'. Argue carefully that Gibbs Sampling (SSS) ill fail here to produce a correct estimate of U. What is it about the transition matrix that describes SSS here that causes this failure? b) Consider the very simple version of the Metropolis-Hastings algorithm here the matrix XœÐ> )) ß Ñ is ) 1(for 1 an ) ) matrix of 's) and ) and ) are vectors for hich the corresponding probabilities in the table above are positive. Find the transition matrix T and verify that it has the distribution above as it invariant distribution. 29. Suppose for sake of argument that I am interested in the properties of the discrete distribution ith probability mass function but, sadly, I don't kno 5. sinb 0ÐBÑ œ 5 for œ Bœßßá B otherise a) Describe a MCMC method I could use to approximate the mean of this distribution. (b) One doesn't really have to resort to MCMC here. One could generate iid observations from 0ÐBÑ (instead of a MC ith stationary distribution 0ÐBÑ) using the rejection method. (And then rely on the LLN to produce an approximate mean for the distribution.) Prove that the folloing algorithm (based on iid Uniform ÐßÑ random variables and samples from a geometric distribution) ill produce from the this distribution. Step 1 Generate from the geometric distribution ith probability mass function Bœßßá 2ÐBÑ œ for œ B otherise Step 2 Generate Y that is Uniform ÐßÑ sinb sinb Step 3 Note that 2ÐBÑ B. If Y2ÐBÑ B set œ. Otherise return to Step 1.

9 30. Suppose that f is a convex subset of Òß_Ñ 5. Argue that there exists a decision problem that has f as its set of risk vectors. (Consider problems here is degenerate and carries no information.) 31. Consider the to state decision problem œöß, T the Bernoulli ˆ % distribution and T the Bernoulli ˆ distribution, T and PÐ) ß+Ñ œmò) Á+ÓÞ a) Find the set of risk vectors for the four nonrandomized decision rules. Plot these in the plane. Sketch f, the risk set for this problem. b) For this problem, sho explicitly that any element of W has a corresponding element of W ith identical risk vector and vice versa. c) Identify the set of all admissible risk vectors for this problem. Is there a minimal complete class for this decision problem? If there is one, hat is it? d) For each : ÒßÓ, identify those risk vectors that are Bayes versus the prior 1 œð:ß :Ñ. For hich priors are there more than one Bayes rule? e) Verify directly that the prescription choose an action that minimizes the posterior expected loss produces a Bayes rule versus the prior 1 œð ß ÑÞ 32. Consider a to state decision problem œöß, here the observable œð ßÑ has iid Bernoulli ˆ % coordinates if ) œ and iid Bernoulli ˆ coordinates if ) œ. Suppose that T and PÐ) ß+Ñ œmò) Á+Ó. Consider the behavioral decision rule 9 B defined by 9 B ÐÖ Ñ œ if B œ, and 9BÐÖ Ñ œ and 9BÐÖ Ñ œ if B œ. a) Sho that 9 B is inadmissible by finding a rule ith a better risk function. (It may be helpful to figure out hat the risk set is for this problem, in a manner similar to hat you did in problem 31.) b) Use the construction from Result 61 and find a behavioral decision rule that is a function of the sufficient statistic and is risk equivalent to 9 B Þ (Note that this rule is inadmissible.) 33. Consider the squared error loss estimation of : ÐßÑ, based on µ binomial B Ðß:Ñ, and the to nonrandomized decision rules $ ÐBÑ œ and $ ÐBÑ œ ˆ B Þ Let < be a randomized decision function that chooses $ ith probability and $ ith probability Þ a) Write out expressions for the risk functions of $, $ and <. b) Find a behavioral rule that is risk equivalent to <. c) Identify a nonrandomized estimator that is strictly better than < or Suppose œ and that a decision rule 9 is such that for each ) ß9is admissible hen the parameter space Ö). Sho that 9 is then admissible hen the parameter space 9

10 35. Suppose that AÐ) Ñ a) Þ Sho that 9 is admissible ith loss function PÐ) ß+Ñ iff it is admissible ith loss function AÐ) ÑPÐ) ß+Ñ. 36. Problem 9, page 209 of Schervish. 10

11 Assignment (Ferguson) Prove or give a counterexample: If V and V are complete classes of decision rules, then V V is essentially complete. 3. (Ferguson and others) Suppose that µ binomial Ðß:Ñ and one ishes to estimate : ÐßÑ. Suppose first that PÐ:ß+Ñ œ: Ð :Ñ Ð: +Ñ Þ a) Sho that is Bayes versus the uniform prior on ÐßÑ. b) Argue that is admissible in this decision problem. c) Sho that is minimax and identify a least favorable prior No consider ordinary squared error loss, PÐ:ß+Ñ œð: +Ñ. d) Apply the result of problem 35 and prove that is admissible under this loss function as ell. 39. Consider a to state decision problem œ T œöß, T0 and T have respective densities ith respect to a dominating 5 -finite measure., 0 and 0 and the loss function is PÐ) ß+Ñ. a) For K an arbitrary prior distribution, find a formal Bayes rule versus KÞ b) Specialize your result from a) to the case here PÐ) ß+Ñ œmò) Á+Ó. What connection does the form of these rules have to the theory of simple versus simple hypothesis testing? 40. Suppose that µ Bernoulli Ð:Ñ and that one ishes to estimate : ith loss $ PÐ:ß+Ñ œl: +l. Consider the estimator $ ith $ ÐÑ œ % and $ ÐÑ œ %. a) Write out the risk function for $ and sho that VÐ:ß$ ÑŸ % Þ b) Sho that there is a prior distribution placing all its mass on Öß ß against hich $ is Bayes. c) Prove that $ is minimax in this problem and identify a least favorable prior. 41. Consider a decision problem here T) is the Normal Ð) ßÑ distribution on k= e, T œöß and PÐ) ß+Ñ œmò+ œómò) &Ó MÒ+ œómò) Ÿ&Ó. a) œð _ß&Ó Ò'ß_Ñ guess hat prior distribution is least favorable, find the corresponding Bayes decision rule and prove that it is minimax. b) œ e, guess hat decision rule might be minimax, find its risk function and prove that it is minimax. 42. (Ferguson and others) Consider (inverse mean) eighted squared error loss estimation of -, the mean of a Poisson distribution. That is, let A œðß_ñß T œ AßT - be the Poisson distribution on k œößßß$ßþþþ and PÐ-ß+Ñ œ - Ð- +Ñ ÞLet $ÐÑ œþ a) Sho that $ is an equalizer rule. b) Sho that $ is generalized Bayes versus Lebesgue measure on A. c) Find the Bayes estimators rt the > Ð, Ñ priors on A. d) Prove that $ is minimax for this problem. 11

12 43. Problem 14, page 340 Schervish. 44. Let RœÐR ßRßÞÞÞßRÑ be multinomial Ðß:ß:ßÞÞÞß:Ñ here : œþ a) Find the form of the Fisher information matrix based on the parameter :. b) Suppose that :Ð 3 ) Ñ for 3œßßÞÞÞß5are differentiable functions of a real parameter and open interval, here each :Ð) Ñ and :Ð) ÑœÞ 3 3 Suppose that 2ÐC ßCßÞÞÞßCÑ 5 is a continuous real-valued function ith continuous first partial derivatives and define ;Ð) Ñœ2Ð: Ð) Ñß:Ð ) ÑßÞÞÞß:ÐÑÑ 5 ). Sho that the information bound for unbiased estimators of ;Ð) Ñin this context is Ð; Ð) ÑÑ. here MÐ ) Ñœ :Ð 3 ) Ñ Œ log:ð 3 ) Ñ Þ M Ð) Ñ.) 5 3œ 45. Sho that the C-R inequality is not changed by a smooth reparameterization. That is, suppose that c œöt ) is dominated by a 5 -finite measure. and satisfies i) 0ÐBÑ ) for all ) and B,. ii) for all B,.) 0ÐBÑ ) exists and is finite everyhere e and iii) for any statistic $ ith E ) l$ ÐÑl _ for all ), E ) $ ÐÑ can be differentiated under the integral sign at all points Let 2 be a function to e such that 2 is continuous and nonvanishing Let ( œ2ð) Ñ and define U( œt). Sho that the information inequality bound obtained from ÖU ( evaluated at ( œ2ð) Ñis the same as the bound obtained from c. 46. As an application of Lemma (or ) consider the folloing. Suppose that ) ) µ U Ð) ß) Ñ and consider unbiased estimators of ) Ð ß) Ñœ. For ) ) ) 0 Ð) ß ÑÐBÑ 0Ð ß ÐBÑ ) ) ) Ñ 0 ÐBÑ 0 ÐBÑ and ) ) ), apply Lemma ith 1ÐBÑ œ and 1ÐBÑ œ Þ Ð) ß) Ñ Ð) ß) Ñ What does Lemma then say about the variance of an unbiased estimator of ) ( ß) Ñ? (If you can do it, sup over values of ) and ). I've not tried this, so am not sure ho it comes out.) 47. Problem 11, page 39 Schervish. 4. Consider the estimation problem ith iid T) observations, here T) is the exponential distribution ith mean ). Let Z be the group of scale transformations on k œðß_ñ ß Z œö1-l- here 1ÐBÑ - œ-bþ a) Sho that the estimation problem ith loss PÐ) ß+Ñ œðlog Ð+Î) ÑÑ is invariant under Z and say hat relationship any equivariant nonrandomized decision rule must satisfy. b) Sho that the estimation problem ith loss PÐ) ß+Ñ œð) +Ñ Î) is invariant under Z, and say hat relationship any equivariant nonrandomized estimator must satisfy. 12

13 c) Find the generalized Bayes estimator of ) in situation b) if the prior has density rt Lebesgue measure 1Ð) Ñœ ) MÒ) Ó. Argue that this estimator is the best equivariant estimator in the situation of b). 13

14 Assignment Problem 19, page 341 Schervish. Also, find the MRE estimator of ) under squared error loss for this model. 50. Let 0Ð?ß@Ñ be the bivariate probability density of the distribution uniform on the square in Ð?ß@Ñ-space ith corners at Ð ÈßÑßÐß ÈÑßÐ ÈßÑand Ðß ÈÑ. Suppose that œð ßÑ has bivariate probability density 0ÐBl) Ñœ0ÐB ) ßB ) Ñ. Find explicitly the best location equivariant estimator of ) under squared error loss. (It may ell help you to visualize this problem to sketch the joint density here for ) œ(.) 51. Consider again the scenario of problem 31. There you sketched the risk set f. Here sketch the corresponding set i. 52. Prove the folloing filling-in lemma: Lemma Suppose that 1 and 1 are to distinct, positive probability densities defined on an interval in e Þ If the ratio 1Î1 is nondecreasing in a real-valued function XÐBÑ, then the family of densities Ö1 ÒßÓ for 1 œ 1 Ð Ñ1 has the MLR property in XÐBÑ. 53. Consider the to distributions T and T on k œðßñ ith densities rt Lebesgue measure 0ÐBÑ œ and 0ÐBÑ œ$b. a) Find a most poerful level œþ test of H ÀµT vs H ÀµT. b) Plot both the - loss risk set f and the set i œö 9 Ð) Ñß) Ð Ñ for the simple versus testing problem involving 0 and 0. What test is Bayes versus a uniform prior? This test is best of hat size? c) Take the result of Problem 52 as given. Consider the family of mixture distributions c œöt ) here for ) ÒßÓ, T) œð ) ÑT ) T. In this family find a UMP level œþ test of the hypothesis H À ) ŸÞ& vs H À ) Þ& based on a single observation. Argue carefully that your test is really UMP. 54. Consider a composite versus composite testing problem in a family of distributions c œöt ) dominated by the 5 -finite measure., and specifically a Bayesian decisiontheoretic approach to this problem ith prior distribution K under - loss. a) Sho that if KÐ@ Ñœ then 9 ÐBÑ» is Bayes, hile if KÐ@ 1Ñœ then 9ÐBÑ»0is Bayes. b) Sho that if KÐ@ Ñ and KÐ@ 1Ñ then the Bayes test against K has Neyman-Pearson form for densities 1 and 1 on defined by k 14

15 1ÐBÑ œ ' 0ÐBÑ.KÐ) Ñ ' 0ÐBÑ.KÐ) Ñ 1 and 1ÐBÑ 1 œ. KÐ@ Ñ KÐ@ ) 1 c) Suppose that is Normal (),1). Find Bayes tests of H À ) œ vs H À ) Á i) supposing that K is the Normal Ðß5 Ñ distribution, and ii) supposing that K is ÐR? Ñ, for R the Normal Ðß5 Ñ distribution, and? a point mass distribution at Fix ÐßÞ&Ñ and - Ð ß Ñ. œö ÒßÓ and consider the discrete distributions ith probability mass functions as belo. B ) œ ) Á )- ˆ - ˆ ˆ - ˆ - ˆ a ) b- Find the size likelihood ratio test of H À ) œ vs H À ) Á. Sho that the test 9 ÐBÑ œmòb œó is of size and is strictly more poerful that the LRT hatever be ). (This simple example shos that LRTs need not necessarily be in any sense optimal.) 56. Suppose that is Poisson ith mean ). Find the UMP test of size œþ& of H À ) Ÿ% or ) 10 vsh À% ). The folloing table of Poisson probabilities TÒ ) œbó ill be useful in this enterprise. B $ % & ' ( ) * ) œ% Þ)$ Þ($$ Þ%'& Þ*&% Þ*&% Þ&'$ Þ% Þ&*& Þ*) Þ$ Þ&$ Þ* Þ' ) œ Þ Þ& Þ$ Þ(' Þ)* Þ$() Þ'$ Þ* Þ' Þ& Þ& Þ$( Þ*%) B $ % & ' ( ) * $ % & ) œ% Þ Þ Þ ) œ Þ(* Þ& Þ$%( Þ( Þ) Þ( Þ$( Þ* Þ* Þ% Þ Þ Þ 57. Suppose that is exponential ith mean - Þ Set up the to equations that ill have to be solved simultaneously in order to find UMP size test of H À - ŸÞ& or - 2 vs H À - ÐÞ&ßÑ. 5. Consider the situation of problem 23. Find the form of UMP tests of H À ) Ÿ ) vs H À ) ) Þ Discuss ho you ould go about choosing an appropriate constant to produce a test of approximately size. Discuss ho you ould go about finding an optimal (approximately) 90% confidence set for ). 59. (Problem 37, page 122 TSH.) Let ßÞÞÞß and ]ßÞÞÞß] 7 be independent samples from N ÐßÑ 0 and N Ð( ßÑ, and consider the hypotheses H À ( Ÿ 0 and H À ( 0Þ Sho that there exists a UMP test of size, and it rejects H hen ] is too large. (If 0 ( is a particular alternative, the distribution assigning probability 1 to the point ith ( œ 0 œð70 ( ÑÎÐ7 Ñ is least favorable at size.) 60. Problem 57 Schervish, page