INFERENCE STATISTICS

Mean and Variance in For members of the exponential families, a special relationship exists between the mean and the variance: the latter is a precisely defined and unique function of the former for each member (Tweedie, 194?1). e The relationship can be shown in the following way. For any likelihood function, LUSH, dcyf) = f(y;; 9;, O and 6 > 0. The probability density function for each Y; is therefore _. i. _ egg; 9 > .3 Suppose .8 is known. f(yla,.3) { 0' elsewhere (3) Show that this distribution belongs to the exponential family. (5) (b) Find the score statistics U and verify that E(U) = 0. (3) (c) Find the information 3 = var[U). (3) (d) Find the maximum likelihood estimator of a. (5) (e) Find the Wald statistic for making inferences about 0:. (4) (f) Use the Wald statistic to obtain an expression for an approximate 95% condence interval for a