Question

1. Suppose that we observe X = (X1,...,xn) consisting of independent and identically exponentially distributed (i.i.d) random variables with parameter (and thus mean 1/0), i.e., X; - Expe), i = 1,...,n, where is completely unknown to us. (a) Describe the state space S of X and the parameter space [1 mark]. (b) Write down the p.d.f. of X and the corresponding statistical model [2 mark]. (c) Prove that T(X) = Ï X, is a sufficient statistic for 0: (i) using the first theorem for sufficient statistics; [3 marks] (Hint: Sums of exponential random variables follow the Gamma distribution] (ii) using the Fisher-Neymann factorization theorem. (1 mark] (d) Show that T(X) = Ï Xi is a minimal sufficient statistic for 0. [2 marks] (e) Show that T2(X) = X = 1 X; is also a sufficient statistic for 0. [1 mark] i=1