Stat 643 Problems. a) Show that V is a sub 5-algebra of T. b) Let \ be an integrable random variable. Find EÐ\lVÑÞ

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1 Stat 643 Problems 1. Let ( HT, ) be a measurable space. Suppose that., T and Uare 5-finite positive measures on this space with T Uand U...T.T.U a) Show that T. and œ a.s......u.....u.u Š.. b) Show 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) Show that V is a sub 5-algebra of T. b) Let \ be an integrable random variable. Find EÐ\lVÞ 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 ß and. 2-dimensional Lebesgue measure. Consider the variable \Ð= œ = and the sub 5-algebra of T generated by \, V. a) For E T, find E ÐMElVœTÐElV. b) For ]Ð= œ =, find E Ð]lV. 4. Prove Result Prove the fact stated in the paragraph above Result 5 in the typed notes. 6. Show directly that the following families of distributions have countable equivalent subsets: a) c, for T) uniform on the integers ) through œöþþþß ß ßßßß$ßÞÞÞ. b) c, for T ) uniform on the 2-dimensional disk of radius ) centered at the œðß_. c) The Poisson Ð) œðß_. 7. Suppose that \œð\ ß\ßÞÞÞß\ has independent components, where each \ 3 is generated as follows. For independent random variables [ 3 µ normal Ð. ß and ^ 3 µ Poisson Ð., \ 3 œ[ 3 with probability : and \ 3 œ^ 3 with probability :. Suppose that. Òß_. Use the factorization theorem and find low dimensional sufficient statistics in the cases that: a) : is known to be, and b) : ÒßÓ is unknown. (In the first case the parameter space œö Òß_, while in the second it is ÒßÓ Òß_.). (Ferguson) Consider the probability distributions on Ðe ßU defined as follows. For ) œð) œ e Ðß and \µt, suppose ) -1-

2 TÒ\ œbó œ ) œ B ) Ð :: for Bœ )) ß ß) ßÞÞÞ otherwise. Let \ß\ßÞÞÞß\ 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 XÐ\ œð min \ß\is 3 3 sufficient for the parameter ). c) Argue that the factorization theorem can be applied if ) is known (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 known, min \ 3 is a sufficient statistic. 9. Suppose that \ w is exponential with mean - (i.e. has density 0ÐB - œ -exp Ð -BMÒB Ó with respect to Lebesgue measure on e, but that one w w only observes \œ\mò\ Ó. (There is interval censoring below Bœ.) a) Consider the measure. on k œö Ðß_ consisting of a point mass of 1 at 0 plus Lebesgue measure on (1,_). Give a formula for the R-N derivative of \ T - wrt. on k. \ b) Suppose that \ß\ßÞÞÞß\ are iid with 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. 10. Suppose that \ß\ßÞÞÞß\ are iid with marginal density with respect to Lebesgue measure on Ðß of the form B 0ÐB ) œœ ) Ð ) B Ðß B Òß Þ Find a low dimensional sufficient statistic and prove that it is minimal. 11. (Lehmann TPE page 65) Let \ ß\ßÞÞÞß\ 7 and ]ß]ßÞÞÞß] be independent normal random variables with E\œ 0ß Var \œ 5, E ] œ ( and Var ] œ 7. Find minimal sufficient statistics for the following situations: a) no restrictions on the parameters, b) 5 œ 7, and c) 0 œ (. 12. (Lehmann TPE page 65) Let 0 be a positive integrable function over Ðß_ and let :ÐB ) œ-ð) 0ÐBMÒ B ) ÓÞ If \ ß\ßÞÞÞß\ are iid with the marginal density :ÐB ), show that max \ 3 is sufficient for ). 13. Schervish Exercise 2.. And show that ^ is ancillary. -2-

3 14. Schervish Exercise (I think you need to assume that ÖB spans e.) 15. Schervish Exercise Schervish Exercise Schervish Exercise What is a first order ancillary statistic here? 1. Schervish Exercise Suppose that f is a convex subset of Òß_ 5. Argue that there exists a decision problem that has f as its set of risk vectors. (Consider problems where \ is degenerate and carries no information.) 20. Consider the two state decision problem œöß, T the Bernoulli ˆ % distribution and T the Bernoulli ˆ distribution, T and PÐ) ß+ 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, show explicitly that any element of W has a corresponding element of W with identical risk vector and vice versa. c) Identify the the set of all admissible risk vectors for this problem. Is there a minimal complete class for this decision problem? If there is one, what is it? d) For each : ÒßÓ, identify those risk vectors that are Bayes versus the prior 1 œð:ß :. For which 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 Þ 21. Consider a two state decision problem œöß, where the observable \œð\ ß\ has iid Bernoulli ˆ % coordinates if ) œ and iid Bernoulli ˆ coordinates if ) œ. Suppose that T and PÐ) ß+ œmò) Á+Ó. Consider the behavioral decision rule 9 B defined by 9 B ÐÖ œ if B œ, and 9BÐÖ œ and 9BÐÖ œ if B œ. a) Show that 9 B is inadmissible by finding a rule with a better risk function. (It may be helpful to figure out what the risk set is for this problem, in a manner similar to what you did in problem 20.) b) Use the construction from Result 21 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.) 22. Consider the squared error loss estimation of : Ðß, based on \µ binomial B Ðß:, and the two nonrandomized decision rules $ ÐB œ and $ ÐB œ ˆ B Þ Let -3-

4 < be a randomized decision function that chooses $ with probability and $ with 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 when the parameter space Ö). Show that 9 is then admissible when the parameter space 24. Suppose that AÐ) a) Þ Show that 9 is admissible with loss function PÐ) ß+ iff it is admissible with loss function AÐ) PÐ) ß Let T and T be two distributions on k and 0 and 0 be densities of these with respect to some dominating 5 -finite measure.. Consider the parametric family of distributions with parameter : ÒßÓ and densities wrt. of the form 0ÐB œð :0 ÐB :0ÐB ß : and suppose that \ has density 0 :. a) If KÐÖ œkðð1} œ, find the squared error loss formal Bayes estimator of : versus the prior KÞ b) Suppose now that K is the uniform distribution on ÒßÓ. Find the squared error loss Bayes estimator of : versus the prior KÞ c) Argue that your estimator from b) is admissible. 26. For the model of problem 25, suppose that T is the binomial Ðß % distribution and 3 T1 is the binomial Ðß % distribution. Find an ancillary statistic. (Hint: consider the probabilities that \ takes the values and.) Is the family of problem 25 then necessarily boundedly complete? 27. (Ferguson) Prove or give a counterexample: If V and V are complete classes of decision rules, then V V is essentially complete. 2. Problem 9, page 209 of Schervish. 29. (Ferguson and others) Suppose that \µ binomial Ðß: and one wishes to estimate : Ðß. Suppose first that PÐ:ß+ œ: Ð : Ð: + Þ \ a) Show that is Bayes versus the uniform prior on Ðß. \ b) Argue that is admissible in this decision problem. Now consider ordinary squared error loss, PÐ:ß+ œð: +. \ c) Apply the result of problem 24 and prove that is admissible under this loss function as well. -4-

5 30. Consider a two state decision problem œ T œöß, T0 and T have respective densities with respect to a dominating 5 -finite measure., 0 and 0 and the loss function is PÐ) ß+. a) For K an arbitrary prior distribution, find a formal Bayes rule versus KÞ b) Specialize your result from a) to the case where PÐ) ß+ œmò) Á+Ó. What connection does the form of these rules have to the theory of simple versus simple hypothesis testing? 31. (Ferguson and others) Suppose that one observes independent binomial random variables, \µ binomial Ðß: and ] µ binomial ( ß: and needs to estimate : : with squared error loss, PÐ: ß+ œð: : +. (We will assume that the parameter space is Ðß and that the action space is Ð ß.) Find the form of a rule Bayes against a prior uniform over the parameter space. 32. Consider a linear loss two action decision problem, = e, T œöß and PÐ) ß+ œð) ) MÒ+ œómò ) ) Ó Ð) ) MÒ) œómò) ) Ó for some ) of interest. Find the form of a Bayes rule for this problem. (Hint: Consider the difference PÐ) ß PÐ) ßÞ 33. Problem 24, page 211 Schervish. 34. Problem 34, page 142 Schervish. 35. Problem 36, page 142 Schervish. 36. Problem 2, page 339 Schervish. 37. Problem 3, page 339 Schervish. 3. Problem 4, page 339 Schervish. 39. Problem 5, page 339 Schervish. 40. Problem 7, page 339 Schervish. 41. Consider the family of bivariate observations \œð]ß^ on k œðß with densities wrt two-dimensional Lebesgue measure 0ÐB ( º exp Ð( C ( D for a bivariate parameter vector ( œð( ß(. a) Find the natural parameter space for this family of distributions. b) What is the normalizing constant OÐ( for this family? Use it to find E( ]Þ c) Find a UMVUE of the quantity E( ] E ( ^ based on iid vectors \ß\ßÞÞÞß\ with marginal (bivariate) distribution specified by 0ÐB (. Argue carefully that your estimator is UMVUE. -5-

6 42. Suppose that \µ Bernoulli Ð: and that one wishes to estimate : with loss $ PÐ:ß+ œl: +l. Consider the estimator $ with $ Ð œ % and $ Ð œ %. a) Write out the risk function for $ and show that VÐ:ß$ Ÿ % Þ b) Show that there is a prior distribution placing all its mass on Öß ß against which $ is Bayes. c) Prove that $ is minimax in this problem and identify a least favorable prior. 43. Consider a decision problem where T) is the Normal Ð) ß distribution on k= e, T œöß and PÐ) ß+ œmò+ œómò) &Ó MÒ+ œómò) Ÿ&Ó. a) œð _ß&Ó Ò'ß_ guess what prior distribution is least favorable, find the corresponding Bayes decision rule and prove that it is minimax. b) œ e, guess what decision rule might be minimax, find its risk function and prove that it is minimax. 44. Consider the scenario of problem 29. Show that the estimator there is minimax for the weighted squared error loss and identify a least favorable prior. 45. (Ferguson and others) Consider (inverse mean) weighted squared error loss estimation of -, the mean of a Poisson distribution. That is, let A œðß_ß T œ AßT - be the Poisson distribution on k œößßß$ßþþþ and PÐ-ß+ œ - Ð- + ÞLet $Ð\ œ\þ a) Show that $ is an equalizer rule. b) Show that $ is generalized Bayes versus Lebesgue measure on A. c) Find the Bayes estimators wrt the > Ð, priors on A. d) Prove that $ is minimax for this problem. 46. (Berger page 35) Suppose that \ is Poisson with mean - and that one wishes to test H À - œ vs H a À - œ with loss, i.e. A œ T œöß and PÐ-ß+ œmò- Á+Ó. a) Find the form of the nonrandomized Bayes tests for this problem. b) Using your answer to part a), find enough risk points for nonrandomized Bayes rules to sketch the part of the lower boundary of the risk set relevant to finding a minimax rule. c) On the basis of your sketch from part b), find a minimax rule for this problem and identify a least favorable distribution. 47. Consider the left-truncated Poisson distribution with pmf B 0ÐB ) œ expð )) ÎBxÐ exp Ð ) for Bœßß$ßÞÞÞ. Suppose that \ß\ßÞÞÞß\ are iid variables with this distribution. a) Show that Xœ \ is a complete sufficient statistic for Ðß_Þ 3œ 3 ) b) It turns out that the pmf of X is given by 1Ð> ) œw> x) > Î>xÐexp Ð) for >œß ß ßÞÞÞwhere W is the so called Stirling number of the second > -6-

7 kind (whose exact definition is not important here). Find the UMVUE of ). (It will involve the W > 's.) 4. Consider an observable \ taking values in k and with distribution T) for Þ For a statistic X, let U œ UÐX and T) be the restriction of T) to U. Prove that if X is complete and sufficient and 9 ÐX is unbiased for ) Ð, then another random variable Y is an unbiased estimator of ) Ð iff E ÐYlXœ 9 ÐX a.s. T a). 49. Suppose that under the usual set-up X and W are two real-valued statistics. a) Suppose that E ) ÐXÐ\ _ and E ) ÐWÐ\ _a). Show that if XÐ\ and WÐ\ are both unbiased estimators of ) and XÐ\ and WÐ\ XÐ\ are uncorrelated a), then Var) XÐ\ Ÿ Var) WÐ\Þ b) Suppose that WÐ\ is an unbiased estimator of ), E ) ÐWÐ\ _a) and that Vœ E ) ÒWlXÓ doesn't depend upon ). Argue that V is preferable to W under squared error loss. 50. Suppose that \ is a real-valued random variable with density (wrt the 5-finite measure.) defined by 0ÐB ( œ exp Ð+Ð( ( B2ÐB for a positive function 2Ð ab e and ( Ð-ß. œ > e, the natural parameter space. a) Find ( Ð œ E( \Þ b) For a fixed ( >, let QÐ= ( œ Eexp ( Ð=\Þ Find QÐ= (, determine for which values of = it is finite, and show that it depends on ( only through ( Ð. 51. Problem 14, page 340 Schervish. 52. Problem 19, page 341 Schervish. Also, find the MRE estimator of ) under squared error loss for this model. 53. Show 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) 0ÐB ) for all ) and B,. ii) for all B,.) 0ÐB ) exists and is finite everywhere e and iii) for any statistic $ with E ) l$ Ð\l _ for all ), E ) $ Ð\ can be differentiated under the integral sign at all points w Let h be a function to e such that 2 is continuous and nonvanishing Let ( œ2ð) and define U( œt). Show that the information inequality bound obtained from ÖU ( evaluated at ( œ2ð) is the same as the bound obtained from c. 54. Let RœÐR ßRßÞÞÞßR be multinomial Ðß:ß:ßÞÞÞß: where : œþ a) Find the form of the Fisher information matrix based on the parameter :. b) Suppose that :Ð 3 ) for 3œßßÞÞÞß5are differentiable functions of a real parameter and open interval, where each :Ð) and :Ð) œþ Suppose that 2ÐC ßCßÞÞÞßC 5 ) 3 3 is a continuous real-valued function with continuous -7-

8 first partial derivatives and define ;Ð) œ2ð: Ð) ß:Ð ) ßÞÞÞß:Ð 5 ). Show that the information bound for unbiased estimators of ;Ð) in this context is w Ð; Ð). where MÐ ) œ :Ð 3 ) Œ log:ð 3 ) Þ M Ð).) 5 w 55. As an application of Lemma 61 (or 61 ) consider the following. Suppose that ) ) w \µ U Ð) ß) and consider unbiased estimators of ) Ð ß) œ. For ) ) ) 0 w Ð) w ß ÐB 0Ð ß w ÐB ) ) ) 0 ÐB 0 ÐB and ) ) ), apply Lemma 61 with 1ÐB œ and 1ÐB œ Þ 3œ Ð) ß) Ð) ß) What does Lemma 61 then say about the variance of an unbiased estimator of ) ( ß)? w w (If you can do it, sup over values of ) and ). I've not tried this, so am not sure how it comes out.) 56. Problem 11, page 39 Schervish. 57. Take the simplest possible -dimensional statistical estimation problem, \ß\ßÞÞÞß\ iid Bernoulli Ð:, and make your way through the results on maximum likelihood, carefully seeing and saying what they assume and what they yield. (Find D Ð:, 6ÐB ß:, verify hypotheses of the theorems and see what they say about : estimation of :Þ For sake of argument, let X œ & \ 3 and have a look at the 3œ corresponding :. What do the theorems say about the behavior of \ when :œ or? What are approximate confidence intervals for :? Etc.) 5. Suppose that \ß\ßÞÞÞ are iid, each taking values in k œößß with R-N derivative of wrt to counting measure on k T ) 0ÐB œ ) & expðb) exp) expð ) 3 3œ a) Find an estimator of ) based on œ MÒ\ œó that is È consistent (i.e. for which ÈÐs ) ) converges in distribution). b) Find in more or less explicit form a one step Newton modification of your estimator from a). c) Prove directly that your estimator from b) is asymptotically normal with variance ÎM Ð). (With ) ^ the estimator from a) and ) the estimator from b), w ) ) œs PÐs ), PÐ ww s) and write PÐs œpð Ðs ) ) ) ) P Ð ) Ðs) ) P Ð) for some ) s w w ww www between ) and ).) d) Show that provided B Ðß the loglikelihood has a maximizer. --

9 B $B 'B )s œ log È Þ Ð B Prove that an estimator defined to be )s when B Ðß will be asymptotically normal with variance ÎM Ð). e) Show that the observed information and expected information approximations lead to the same large sample confidence intervals for ). What do these look like based on, say,? )s By the way, a version of nearly everything in this problem works in any one parameter exponential family. See the Theory II question on last year's prelim. 59. Prove the following filling-in lemma: Lemma Suppose that 1 and 1 are two distinct, positive probability densities defined on an interval in e Þ If the ratio 1Î1 is nondecreasing in a real-valued function XÐB, then the family of densities Ö1 ÒßÓ for 1 œ 1 Ð 1 has the MLR property in XÐB. 60. Consider the situation of problem 9 and maximum likelihood estimation of -. a) Show that with Qœ Ò\ 3's equal to Ó, in the event that Qœthere is no MLE of -, but that in all other cases there is a maximizer of the likelihood. Then argue that for any -, with T probability tending to 1, the MLE of -, say - ^ -, exists. b) Give a simple estimator of - based on Q alone. Prove that this estimator is consistent for -. Then write down an explicit one-step Newton modification of your estimator from a). c) Discuss what numerical methods you could use to find the MLE from a) in the event that it exists. d) Give two forms of large sample confidence intervals for - based on the MLE - ^ and two different approximations to MÐ-Þ 61. Consider the two distributions T and T on k œðß with densities wrt Lebesgue measure 0ÐB œ and 0ÐB œ$b. a) Find a most powerful level œþ test of H À\µT vs H À\µT. b) Plot both the - loss risk set f and the set i œö 9 Ð) ß) Ð for the simple versus testing problem involving 0 and 0. What test is Bayes versus a uniform prior? This test is best of what size? c) Take the result of Problem 59 as given. Consider the family of mixture distributions c œöt ) where for ) ÒßÓ, T) œð ) T ) 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. -9-

10 62. 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 with prior distribution K under - loss. a) Show that if KÐ@ œ then 9 ÐB» is Bayes, while if KÐ@ 1œ then 9ÐB»0is Bayes. b) Show that if KÐ@ and KÐ@ 1 then the Bayes test against K has Neyman-Pearson form for densities 1 and 1 on defined by 1ÐB œ ' k 0ÐB.KÐ) ' 0ÐB.KÐ) 1 and 1ÐB 1 œ. KÐ@ ) 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 distribution, and ii) supposing that K is ÐR?, for R the Normal Ðß5 distribution, and? a point mass distribution at Suppose that \ is Poisson with mean ). Find the UMP test of size œþ& of H À ) Ÿ% or ) 10 vsh À% ). The following table of Poisson probabilities TÒ\ ) œbó will 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 64. Suppose that \ is exponential with mean - Þ Set up the two equations that will have to be solved simultaneously in order to find UMP size test of H À - ŸÞ& or - 2 vs H À - ÐÞ&ß. 65. Suppose that for 3œßßÞÞÞ the random vectors \ 3 œð] 3 ß^ 3 are iid with distribution T : described as follows. ] and ^ are independent, ] µ binomial Ðß: D and ^ µ geometric Ð: (i.e. with pmf 1ÐD : œ:ð : for DœßßÞÞÞ). a) Give the C-R lower bound on the variance of an unbiased estimator of : based on \ ß \ ßÞÞÞß\ Þ b) Find the maximum likelihood estimator of : based on \ ß \ ßÞÞÞß \ and discuss its asymptotic properties. c) Give both the expected information and observed information large sample confidence intervals for : based on your estimator from b). 66. Consider again the scenario of problem 20. There you sketched the risk set f. Here sketch the corresponding set i

11 67. Consider the estimation problem with iid T) observations, where T) is the exponential distribution with mean ). Let Z be the group of scale transformations on k œðß_ ß Z œö1-l- where 1ÐB - œ-bþ a) Show that the estimation problem with loss PÐ) ß+ œðlog Ð+Î) is invariant under Z and say what relationship any equivariant nonrandomized decision rule must satisfy. b) Show that the estimation problem with loss PÐ) ß+ œð) + Î) is invariant under Z, and say what relationship any equivariant nonrandomized estimator must satisfy. c) Find the generalized Bayes estimator of ) in situation b) if the prior has density wrt Lebesgue measure 1Ð) œ ) MÒ) Ó. Argue that this estimator is the best equivariant estimator in the situation of b). 6. (Problem 37, page 122 TSH.) Let \ßÞÞÞß\ and ]ßÞÞÞß] 7 be independent samples from N Ðß 0 and N Ð( ß, and consider the hypotheses H À ( Ÿ 0 and H À ( 0Þ Show that there exists a UMP test of size, and it rejects H when ] \ is too large. (If 0 ( is a particular alternative, the distribution assigning probability 1 to the point with ( œ 0 œð70 ( ÎÐ7 is least favorable at size.) 69. Problem 57 Schervish, page Problem 12, parts a)-c) Schervish, page Fix ÐßÞ& and - Ð ß. œö ÒßÓ and consider the discrete distributions with probability mass functions as below. B ) œ ) Á )- ˆ - ˆ ˆ - ˆ - ˆ a ) b- Find the size likelihood ratio test of H À ) œ vs H À ) Á. Show that the test 9 ÐB œmòb œó is of size and is strictly more powerful that the LRT whatever be ). (This simple example shows that LRTs need not necessarily be in any sense optimal.) 72. Consider the situation of problem 5. a) Find the form of UMP tests of H À ) Ÿ ) vs H À ) ) Þ Discuss how you would go about choosing an appropriate constant to produce a test of approximately size. Discuss how you would go about finding an optimal (approximately) 90% confidence set for ). b) Beginning with a third order Taylor expansion of the loglikelihood at ) about the MLE, prove directly that the asymptotic ; distribution holds under H for the likelihood ratio statistic for testing H À ) œ ) vs H À ) Á) Þ c) Suppose that œ and œ(ß œ' and œ(. Plot the corresponding loglikelihood and give the value of the MLE. Based on the MLE and its normal limiting distribution, then give an approximate 90% two-sided confidence interval -11-

12 for ). Finally, based on the asymptotic ; distribution of the likelihood ratio statistic, give a different approximate 90% confidence interval for ). 73. Consider the situation of Problems 9 and 60. Below 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 w values given above). Give approximate 90% two-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. Now consider the censored data problem where any value less than 1 is reported as 0. Modify the above data accordingly and do the following. b) Plot the loglikelihood for the censored data (the derived B) values. How 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 - MÐ -œ Ð Þ Œ ˆ - exp - - expð - Give two 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 74Þ Suppose that \ß\ß\ $ and \% are independent binomial random variables, \ 3 µ binomial Ðß: 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 show 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 showing the ; limit of the LRT statistic under the null hypothesis). b) Find the form of the Wald tests and show directly that the test statistic is asymptotically ; under the null hypothesis. c) Find the form of the ; tests of these hypotheses and again show directly that the test statistic is asymptotically ; under the null hypothesis. 75. If \ß\ßÞÞÞß\ are iid normal Ð.5 ß, find the form of the LRT of H À. œ. vs H À. Á. Þ Show directly that the LRT statistic is asymptotically ; with 1 degree of freedom. Invert the LRTs to produce a large sample 90% confidence procedure for