QUESTION 1 [41] A certain company manufactures electronic components daily. Some of the components manufactured by the company can be defective. Let X (0,1) be a random variable representing the daily proportion of defective electronic components manufactured by the company. It has been hypothesized that the probability density function of X is: (B + 1) xif 0 0 is known. The company wants to know B. Let X1, X2, ..., X, be the proportion of defective components manufactured on n randomly chosen days. That is, X1, X2, ..., X, is a random sample from a distribution with probability density function f(xIB). n (a) Use the factorization theorem/criterion to show that II X, is a sufficient statistic for B. (4) i=1 (b) Show that I x; is a minimal sufficient statistic for B. (4) i=1 (C) Show that f (x\B) belongs to the regular 1-parameter exponential family by showing that the probability density function can be expressed as: g(x) exp{0t(x) - YO)} Do not forget to identify: 0; g(x); t(x); and y() in the expression. Given: v'O) = E[t(X)] and y"0) = Var [t(X)]. (6) (d) What is the complete sufficient statistic for B? Justify your answer. (3) (e) What is the mean and the variance of the complete sufficient statistic for B in terms of B? (1) Use part (e) to find a method of moments estimator of B in terms of the complete sufficient statistic for B. X-1 (g) Prove or disprove that B is another method of moments estimator of p. (6) 1-X (h) Find the maximum likelihood estimator of B. (7) 2X B2 (i) Find the maximum likelihood estimator of (3) (n+ 2)(n+ 1)2 Justify your answer. [TURN OVER 4 STA3702 SEP/OCT/NOV/DEC 2021 QUESTION 2 [24] n Refer to QUESTION 1. Let T* = In X). Given: i=1 B = and Var n-1 (n-2)(n-1) (a) Find: (i) the Fisher information for B in the sample: (3) (ii) the observed Fisher information for p in the sample; and (2) (iii) the Cramer-Rao lower bound of the variance of an unbiased estimator of B. (2) (b) Find the minimum variance unbiased estimator of B. Justify your answer. Do not use the Cramer-Rao lower bound approach. (4) (c) Prove or disprove that the maximum likelihood estimator of in QUESTION 1 part (h) is a consistent estimator of B. Hint: Express the maximum likelihood estimator in terms of T* above. (4) (d) Show that the minimum variance unbiased estimator in part (b) is a better estimator in terms of the mean squared error criterion. (5) (e) Write down an expression for the large sample 95% confidence interval for B in terms of the maximum likelihood estimate of B