QUESTION 1

A distribution belongs to the regular 1-parameter exponential family if among other regularity conditions

its pd f or pmf has the form:

Question 1 A distribution belongs to the regular 1-parameter exponential family if among other regularity con- ditions its pdf or pmf has the form: f (x|0) = g (x) exp [0t (x) - v (0)], 0 6 0 c (00, 00). Furthermore, for this distribution E[t (X)] = v'(@) and Var [t (X)] = w"(0). Let X1, X2, X3, ..., Xn be a random sample from a population whose probability density function is f (x |0) = 8+1 if 2 2, otherwise. (a) Show that f (x|0) belongs to the 1-parameter exponential family. (b) What is the complete sufficient statistic for 0. Justify your answer. (c) Find the mean and the variance of the complete sufficient statistic for 0. (d) Determine the Cramer-Rao lower bound for an unbiased estimator of 0. (e) Find the minimum variance unbiased estimator of -. Justify your