INFERENCE STATISTICS

NEED HELP FROM QUESTION F TO QUESTION I

(c) Show that f (x L8) belongs to the regular l-paranieter exponential family by showing that the probability density function can be expressed as: g(X)eXP{9f(x) W050} Do not forget to identify: 9; g(x); {(x); and 1;;(6') in the expression. Given: V09) : E [t(X)] and y/TQ) : Var [t(X)]. (6) (d) What is the complete suicient statistic for ,8? Justify your answer. (3) (e) What is the mean and the variance of the complete suicient statistic for [3 in terms of )3"? (4) (i) Use part (e) to nd a method of moments estimator of )8 in terms of the complete suicient statistic for )3 - (4) . _ 2f 1 . . (g) Prove or disprove that [3' : 1 f is another method of moments estimator of ,8. (6) (h) Find the maximum likelihood estimator of )9 . (7) [92 (i) Find the maximum likelihood estimator of w+mw+w as . Justify your answer. (3) _ . QUESTION 1 A certain company manufactures electronic components daily. Some of the components manufactured by the company can be defective. Let X E (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: (,8+l)x'3 ifO 0 is known. The company wants to know ,8. Let X1, X2, ..., X\" be the proportion of defective components manufactured on n randomly chosen days. That is, X1, X 2, .. ., X\" is a random sample from a distribution with probability density function f (x l)