INFERENCE STATISTICS

QUESTION 1 A distribution belongs to the regular 1-parameter exponential family if among other regularity conditions its pdf or pmf has the form: f(v10) = g(v) explot(y) - y(0)), y ex C (-0o, co) and / E O c (-00, 00); where g(y) > 0, t(y) is a function of y which does not depend on 0, and y (0) is real valued function of ( only. Furthermore, for this distribution, if the first and second derivatives of y (0) exist, then E[t(Y)] = y'(0) and Var[t(Y)] = w"(@). Let X1, X2, ..., X, be the survival times of a random sample of n identical electronic components, and suppose that the survival time of a component has a distribution with probability density function Bax"-le-Bx if0 0 is known and a > 0 is unknown. (a) Use the factorization theorem/criterion to show that X, is a sufficient statistic for a. i=1 (b) Show that I T X; is a minimal sufficient statistic for a. i=1 (c) Show that f (x|a) belongs to the regular 1-parameter exponential family by showing that the probability density function can be expressed as: g(x) exp(0t(x) - y (0)} Do not forget to identify: 0; g(x); t(x); and w (0) in the expression. (d) What is the complete sufficient statistic for a? Justify your answer. (e) What are the mean and variance of the complete sufficient statistic for a? (f) Use part (e) to find a method of moments estimator of a. (g) Prove or disprove that In X, is a minimum variance unbiased estimator (MVUE) of -. Hint: You may use your answers to parts (d) and (e).QUESTION 2 Refer to QUESTION 1. 1 (3} Find the maximum likelihood estimator of -. {I (b) Show that the estimator in part (a) is a consistent estimator of i. Hint: You may Use your answer to QUESTION 1 [1311(2)]. '2 (c) Find: (i) the Fisher information for a in the sample; (ii) the observed Fisher information for a in the sample; and 1 (iii) the CrarnerRao IOWer bound of the variance ofan unbiased estinlator of . a l (it) Use your answers to part {b} and (c) to prove that the maximum Iikehood estimator of in part (a) is a . . . . . 1 also a nnmmurn variance unbiased estimator of __ {I (e) Write doWn an expression for the 95% condence interval for at