QUALIFYING EXAMINATION II IN MATHEMATICAL STATISTICS SATURDAY, MAY 9, Examiners: Drs. K. M. Ramachandran and G. S. Ladde

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1 QUALIFYING EXAMINATION II IN MATHEMATICAL STATISTICS SATURDAY, MAY 9, 2015 Examiners: Drs. K. M. Ramachandran and G. S. Ladde INSTRUCTIONS: a. The quality is more important than the quantity. b. Attempt at least 2 problems from each part totaling at most 5 problems. c. In the absence of any Tables, students are expected to provide the solution of the data oriented problems in the form of the problem solving process. d. Students are also expected to exhibit the reading, writing and problem solving abilities. e. Just the mechanical work with an answer (if any) is not enough to receive the full credit. PART-1 1Þ Prove or disprove the following statements: a. The distribution of statistics is population distribution. b. If the marginal densities of random variables are identical,then they are independent. # # c. Z+<Ð+ \ + \ Ñœ+ Z+<Ð\ Ñ + Z+<Ð\ Ñ #+ + G9@Ð\ ß\ ÑÞ " " # # " " # # " # " # 2. Let \ß\ß\ be random sample drawn from exponentialð Ñpopulation. Find: " # $ " (i) \ Ð"Ñ (ii) \ Ð$Ñ (iii) the sample range. 3. Let \ ß \ ß ÞÞß \ be a sample ß \ œ Ð\ ß \ ß ÞÞß \ Ñ, X Ð\Ñ be a statisticsß 8 \œb ; V be observed sample set, and VÐXÑbe the range of statistics X. Let 0ÐBl )) be the joint pdf of the given the sample, \ " # 8 " # 8 X (a) If \ß\ßÞÞß\ " # 8 are Bernoulli random variables with parameter! ) " and 8 XÐ\Ñ œ \, then show that XÐ\Ñis sufficient statistics. 3œ" 3 (b) If \ß\ßÞÞß\ " # 8 are iid observations from discrete uniform distribution on Ö"ß #ß $ß ÞÞÞß ), ) R and X Ð\Ñ œ max 3 Ö\ " ß \# ß ÞÞß \ 8, then show that XÐ\Ñis sufficient statistics.

2 4. Let \ß\ßÞÞß\ " # 8be a random sample from Binomial Ð8ß:Ñdistribution, where both 8 and : are unknown. (a) By employing the Method of Moment, find estimators for both 8 and :. (b) Discuss a situation in which both 8 and : are unknown. (c) Five realizations of a binomial Ð8ß :Ñ experiment are observed. (i) The First data set is: 16, 18, 22, 25, 27, and (ii) the second data set is: 16, 18, 22, 25, 28. For these data sets, using the method of moment,calculate the estimates for 8 and :. (d) Compare the estimates in Problem 4(c). (e) What kind of conclusions can you draw from Problem 4(d)? 5. Let \ ß \ ß ÞÞß \ be a random sampleß \ œ Ð\ ß \ ß ÞÞß \ Ñ ß \ œ B ; V be observed sample set. Let 0ÐBl )) be the joint pdf of the given the sample, \, and let sup@ 9 PÐ) be an entire parameter set in V. 9 Á 9, let us define -ÐBÑ œ sup PÐ ) (a) Show that: (i)!ÿ- ÐBÑŸ"ßfor all B ;, (ii) As PÐ) lbñincreases, -ÐBÑdecreases, PÐs) 9lBÑ sup@ 91ÐXÐBÑl) Ñ (iii) -ÐBÑ œ PÐs) lbñ, (iv) -ÐBÑ œ sup 1ÐX ÐBÑl) Ñ œ - ÐX ÐBÑÑ, where X Ð\Ñ sufficient statistics and 0ÐBl) ) œ 1ÐX ÐBÑl) Ñ2ÐBÑà (iv) As lbñ decreases, -ÐBÑ increases. (b) Under what condition(s) 9, -Ð\Ñ is called as the Likelihood Ratio Test Statistics? (c) Assuming that -Ð\Ñ is Likelihood Ratio Test statistics, what can you say about : ÖB À -ÐBÑ Ÿ - for any number - satisfying! Ÿ - Ÿ "? " # 8 " # 8 X 8 PART-2 1. Let \ be normal random variable with mean. and variance 5 # ÞLet us further # # assume that. µrð. : ß 5: Ñand its prior distribution is 1 (.)..: and 5: are assumed to be either known or estimated. a) Find the marginal distribution of the given random. b) Define the likelihood function, PÐ) lbñ, where B is a realization of random variable \. c) Find a posterior distribution of ), 1 ()lb). d) Find the predictive distribution of an unknown observation B. #. Assume that all the given statements in Problem #"-Part-2 remain unchanged. Further assume that. œ "!ß 5 œ &ß 5 œ "! and B œ " 00Þ : : a) Find IÒ. lbó and IÒÐ. IÒ. ÓÑ # lbó Þ b) Find the precision of the normal distribution. c) Show that the IÒ. Ó œ IÒIÒ. lbóó and interpret it. d) Show that IÒvar Ð. lbñó œ var(.) var( IÒ. lbó) and interpret it.

3 3. Let :ÐBl) Ñ be a conditional density of a random variable \. The Jeffrey's prior is defined by: 1) ( ) º lmð) Ñl " #. Compute the Jeffrey's prior for ) for the following given a random sample \" ß \# ß ÞÞÞß \ 8 with iid: a) Exponential ("). b) Poisson (-). c) RÐ. ß 5 Ñ : : # %. Under the Bayesian approach, for given \ œ B, the 1 ÐÐ) lb) is posterior distribution of ). For ßthe credible probability of Eis defined by: T Ð) ElBÑ œ 1 ÐÐ) lb ).). E (a) Is there any relationship between the credible set and confidence set? Justify. (b) For a given \ß\ßÞÞß\ " # 8 random sample with Poisson (-) distribution, and - has " " a gamma Ð+ß,Ñprior. The posterior pdf of - is gamma Ð+ XÐBÑßÒ8 Ó Ñ, 8 where XÐ\Ñ œ \. Find the credible set for. 3œ" 3 - (c) In addition to the conditions in (b), assume that: 8œ"! and XÐBÑœ'. Find 90% credible set for -. &. Let \ be a random variable with Cauchy Ð!ß "Ñ distribution 0 ÐBÑ œ. \, " Ð" B Ñ 1 # a. Find: (i) J\ ÐBÑ, " (ii) J\ ÐBÑ(if it exists). b. Is it possible simulate the Cauchy Ð!ß "Ñ? Justify. c. If the answer to question in (b) is "YES", then is it possible to simulate Cauchy ( α" ß Ñ? GOOD LUCK

4 DEPARTMENT OF MATHEMATICS AND STATISTICS QUALIFYING EXAMINATION II IN MATHEMATICAL STATISTICS SATURDAY, SEPTEMBER 24, 2016 Examiners: Drs. K. M. Ramachandran and G. S. Ladde INSTRUCTIONS: a. The quality is more important than the quantity. b. Attempt at least 2 problems from each part totaling at most 5 problems. c. In the absence of any Tables, students are expected to provide the solution of the data oriented problems in the form of the problem solving process. d. Students are also expected to exhibit the reading, writing and problem solving abilities. e. Just the mechanical work with an answer (if any) is not enough to receive the full credit. PART-1 1. Let \ be a real-valued random variable defined on a complete probability space ÐH ß ¹ ß TÑ with singleton set range (R( \) œ ÖB any B R; where R is a set of real numbers). a. Find: a distribution function J \ of \. Justify bþ Find a probability density function 0 \ of \ (if exists). c. Draw the sketches of J\ and 0\ with justifications. d. What conclusion can be drawn from Parts (a), (b) and (c)? Justify. 2. Let \ß\ßÞÞß\ " # 8be a Bernoulli-type random sample drawn from a population # # mean. Ò!ß "Ó and variance 5 Ò!ß "ÓÞ \ and W sample mean and variance of the random sample. a. Find the joint distribution of the random sample ; b. Find: IÒ\ Ó; c. Find: Var Ò\Ó. # # d. Show that IÒW Ó œ 5 ß e. Based on your response in (a)-(d), what conclusions can you draw. Þ

5 3. Let \" ß \# ß ÞÞÞß \ 8ß ÞÞÞ be a sequence of iid random variables with mean ( IÒ\ 8 Ó œ., Z+<Ð\ Ñœ5 # ÑÞShow that: 8. # # # (a) W8 converges in probability to 5 as 8Ä, whenever Z+<ÐW8ÑÄ!Þ (b) Prove or disprove convergence or divergence of W 8. # (c) On the basis of (a) and (b),what types of conclusion can you draw about W and W? 8 4. Let \ß\ßÞÞß\ be a random sample defined in Problem 2 ÞLet XÐ\Ñbe defined " # œ" by: XÐ\Ñ œ \ Þ Prove or disprove the following statements: a. The distribution of XÐ\Ñis an exponential family. b. XÐ\Ñis sufficient statistics. 8 B 3 3œ" c. The Bernoulli MLE of. is.sœ Þ 8 5. Let \ ß \ ß ÞÞß \ be a random sampleß \ œ Ð\ ß \ ß ÞÞß \ Ñ ß \ œ B ; V be observed sample set. Let 0ÐBl )) be the joint pdf of the given the sample \, and let 1) ÐÑbe a prior distribution of ). be an entire parameter set in Á9. - Moreover, let T lbñ œ T Ð H! is true lbñ and T lbñ œ T Ð H " is truelbñþ (a) Define the "action set" with respect to the "Bayesian hypothesis testing problem". (b) Is the "hypothesis test" identical with a "decision rule"? Justify. (c) Define the loss function in the "hypothesis testing problem". (d) Define the rejection region for the "Bayesian hypothesis testing problem". (e) Determine the costs for Type I and II Error " # 8 " # 8 X 8 PART-2 1. Let \ be a binomial random variable with mean. Ò!ß "Ó and variance 5 # Ò!ß "Ó, and let \ß\ßÞÞß\ " # 8be a corresponding random sample drawn from this populatiom. In the Bayesian analysis, it is assumed that the parameter. is random variable. Let :ÐCß. Ñ be the joint distribution of Ð\ß. ÑÞ The binomial sampling model is denoted by :ÐCl. ÑÞ a. Justify the exchangeability property of :ÐCß. Ñ. b. Find the expression for the joint probability of \ and. c. Find the marginal distribution of the given random. d. Define the likelihood function, PÐ. lcñ, where B is a realization of random variable \.

6 2. Let ] µ Beta Ðα" ß Ñ and Z µ 0. Further assume Beta Ðα" ß Ñ and 0 have common Z Ò!ß "Ó with α œ #Þ(ß " œ 'Þ$ and - œ 2.67 œ sup 0 ] ÐCÑ Þ Let ÐY ß Z Ñ be independent and uniformly distributed random variables to generate a random variable. Show that: BetaÐα" ß ÑÐZ Ð= ÑÑ BetaÐα" ß ÑÐZ Ð= ÑÑ C (a) T Ö= ÀZÐ= ÑŸCand YÐ= ÑŸ œ -.?.@Þ BetaÐα" ß ÑÐZ Ð= ÑÑ " - - (b) T Ö= À YÐ= Ñ Ÿ = Þ -!! (c) T Ö= À ]Ð= ÑŸC œ T Ö= ÀZÐ= ÑŸC Ö = ÀYÐ= ÑŸ Þ (d) What conclusions can you draw from the results (a)-(c)? Z BetaÐα" ß ÑÐZ Ð= ÑÑ - 3. Let \ be a binomial random variable corresponding to the random sample in Problem 1 (Part 2). In the Bayesian analysis, it is assumed that the parameter. is random variable. Let :ÐCß. Ñ be the joint distribution of Ð\ß. ÑÞ The binomial sampling model is denoted by :ÐCl. ÑÞ In addition assume that :Ð. Ñ is prior distribution.. a. Find a posterior distribution of., :(. lc). b. If. µ Beta ( α, "), the find :(. lc). c. What conclusions can you draw from the conclusion of (b)? d. Using (b) find IÒ. lcó and Var Ð. lcñ. " C C e. Show that IÒ. lcó VarÐ. lcñ Ð" Ñthe α -level credible interval for Prove or disprove the following statement. a. The normal distribution has conjugate prior distribution. b. The Binomial distribution Ð8ß. Ñ has no conjugate prior distribution. c. The Poisson distribution has conjugate prior distribution. 5. Assume that all the conditions in Problem 1 (Part 2) are valid. a. Is it possible to determine the Fisher information MÐ. Ñ,. V? Justify. b. If the answer to the question in (a) is "YES", then find the Jeffrey's prior density. c. Based on your work in (b), is the prior in (b) proper or improper? Justify. GOOD LUCK

7 DEPARTMENT OF MATHEMATICS AND STATISTICS QUALIFYING EXAMINATION II IN MATHEMATICAL STATISTICS SATURDAY, SEPTEMBER 26, 2015 Examiners: Drs. K. M. Ramachandran and G. S. Ladde INSTRUCTIONS: a. The quality is more important than the quantity. b. Attempt at least 2 problems from each part totaling at most 5 problems. c. In the absence of any Tables, students are expected to provide the solution of the data oriented problems in the form of the problem solving process. d. Students are also expected to exhibit the reading, writing and problem solving abilities. e. Just the mechanical work with an answer (if any) is not enough to receive the full credit. PART-1 1. Let \ be a real-valued random variable defined on a complete probability space ÐH ß ¹ ß TÑ with singleton set range (R( \) œ ÖB for some B R). (a) Find: a distribution function J \ of \. Justify (b) Find a probability density function 0 \ of \ (if exists). (c) Draw the sketches of J\ and 0\ with justifications. (d) What conclusion can be drawn from Parts (a), (b), and (c)? Justify. 2. Let \" and \# be two random variables. (a) Does the independence of \" and \# imply J\ œ J\? Justify. " # (b) Does J\ œ J \ imply independence of \" and \#? Justify. " # (c) Under what conditions \ß\ is a random sample of size #? Justify. " # 3. Let \ß\ß\ß\ " # $ % and \& be iid observed from a distribution with mean. and # \ \ " # \ \ \ $ % & variance 5 Þ Let." œ # ß.# œ $ and. 3 œ \ be three estimators of.. (a) Calculate the variance of each estimator. Justify. (b) Compare the estimators in Problem 4 (a). Justify. (c) What conclusions can you draw from Problem 4 (b)? Justify.

8 4. Let H!, H 9 are defined in a hypothesis testing problem. Let \ß\ßÞÞß " # # # \ 8 be a random sample from R(.5 ß ), 5 known, and let \œð\ß\ßþþß " # X 8 \Ñß\œB 8 ; V, and we consider testing : H!:. œ.! versus H 1 :. Á. ). For fixed α level, a reasonable level test has rejection region: ÖlB.! l DαÎ# 5Î 8 Þ Determine: (a) The accepted set of sample elements of ;. (b) The probability of acceptance region. (c) The Confidence interval. 5. Which of the following statements are TRUE or FALSE? Justify your responses. (a) The true value is guaranteed by the point estimator rather than interval estimator. (b) For level α test, TÐReject H! Ÿ α and TÐAccept H! " α provide a basis to determine the α level region parameter estimationþ (c) For +,, the coverage probability of Ò+]ß,]Ó dependents on parameter ). (d) For +,, the coverage probability of Ò+ ]ß, ]Ó is independent of ). PART-2 1. Let \ be normal random variable with mean. and variance 5 # ÞLet us further # # assume that. µrð. : ß 5: Ñand its prior distribution is 1 (.)..: and 5: are assumed to be either known or estimated. a) Find the marginal distribution of the given random. b) Define the likelihood function, PÐ) lbñ, where B is a realization of random variable \. c) Find a posterior distribution of ), 1 ()lb). d) Find the α -level credible interval.. 2. Assume that all the given statements in Problem #4 remain unchanged. Further assume that. œ 3 ß 5 œ 5ß 5 œ 2 and B œ " 20Þ : : a) Find IÒ. lbó and IÒÐ. IÒ. ÓÑ # lbó Þ b) Find the precision of the normal distribution. c) Show that the IÒ. Ó œ IÒIÒ. lbóó and interpret it. d) Show that IÒvar Ð. lbñó œ var(.) var( IÒ. lbó) and interpret it.

9 3. Let :ÐBl) Ñ be a conditional density of a random variable \. The Fisher information # ` " and Jeffrey's prior are defined by: MÐ) Ñ œ IÒ ` # 68Ð:ÐBl) ÑÑ and 1 ()) º lmð) Ñl # ), respectively. Compute the Fisher and Jeffrey's prior for ) for the following given a random sample \" ß \# ß ÞÞÞß \ 8 with iid: a) Binomial(1, )). b) Exponential ()). c) Poisson ()) 4. Let \" ß \# ß ÞÞÞß \ 8be a random sample of size 8 from a population RÐ. ß 5 # Ñ with 5 # œ %Þ Further assume that. µ RÐ!ß "ÑÞ # (a) Find IÒ. lbó and IÒÐ. IÒ. lbóñ Þ (b) Find the precision of the normal distribution. (c) Find *&% credible interval for. 5. A change in stock price of AE follows binomial distribution with parameter! ) "ÞA random sample of 6 price changes were recorder. 3 out of 6 price changes were upward direction. Knowing the past knowledge, the values of prior distribution 1) ( ) at )œ!þ%ß!þ&ß!þ'ß and!þ) are 0.1, 0.2, 0.4, and 0.3, respectively. a) For the given values of ), generate a table with columns: ) and values of: (i) prior distribution, (ii) the likelihood function, (iii) the product of prior distribution and likelihood function, and (iv) posterior probability. b) Find the marginal distribution of the given random sample. c) Find the expected value of ). GOOD LUCK

DEPARTMENT OF MATHEMATICS AND STATISTICS QUALIFYING EXAMINATION II IN MATHEMATICAL STATISTICS SATURDAY, SEPTEMBER 24, 2016

DEPARTMENT OF MATHEMATICS AND STATISTICS QUALIFYING EXAMINATION II IN MATHEMATICAL STATISTICS SATURDAY, SEPTEMBER 24, 2016 Examiners: Drs. K. M. Ramachandran and G. S. Ladde INSTRUCTIONS: a. The quality