Question

1 = .*.vk-1e-ta 1. Suppose that X is a random variable with a Gamma-(k, 1) distribution where k > 0 is known, but > 0 is unknown Væ € (0,1), we have f(x) T(k) Let us use 0 = 1/ which is the standard approach, for example in Hogg and Tanis. Calculate the Fisher information I(C). 2. Continuing with problem 1, suppose that X1, ..., Xn are IID copies of the random variable X. Suppose we take the MLE for based on these random variables 0 = T(X1, ..., Xn) Xi +...+X nk Prove that this attains the lower bound for the variance given by the Cramer-Rao theorem. There are different arguments for this (which may be easier or harder). Choose your favorite one (or ones if you wish to do multiple proofs).

Please Do Q2 only..........

Answer 1

Solection: We will proceed step by step for this problem : Firstly X Gamma(,0) X Gamma (K,05 %). where Kyo is known but o is unknown. fx(x) = AK. TR xyo. 1 xk-1 5 O.CO Fisher Information for o for single sample I(0) Inf(050] Joz га? E 202 2 19 E is [20+ { constant + x dn 6 - 03 ED{+ -*}] - F(-) ka -.ERLB for n vid samples from above rannphe population for the parametrie functior me given by: CRLBO) - (40) NI (0) Now consider the MLE for 0, based on ne random samples ô = T(X - _ . Xn) = Tid Gamma (,0) Voor (xi) = 72. (how?) E(xi) : yo? X+X2t- +Xn vk. since, Xi k K-1 X;. Xi -oxi dai TK O DO K+1-1 -oxi Xi dzi By similar way KH-1 Xi KH -oxi.o E(X7) kéke+1) e dai 02 . Var (xi) E(XY) - EP(xi) ktl. ok k.okti KHI K Vi=1(1) n 02 L2 Vor(xi) - Var n24² again a pdf Var [T(x...,xn) (me 3.xi) ne tikai and from C#), 4(0) - E [T[X1, ---Xn)] = nk E(2xi) - l - CRLB (O)= = Var (T(X1, . . , x n )) (Proved 1 K K n. 4/² 1 2K ak