Let X1 X2 Xn be iid from Exponential with mean 1 a Find

Let X1, X2, . . . , Xn be i.i.d. from Exponential() with mean 1/. (a) Find the Fisher information I(). (b) Find the Cramer-Rao lower bound of any unbiased estimators of . (c) What is the asymptotic distribution of the mle ˆ?

Solution

Exponential distribution, f(x)=*exp(-X) (a):Fisher Neyman Criterian L=^(n)*exp(-summation over i=1 to n(*Xi) taking logarithms, logL=n*log-*(summation over i=1 to n(Xi)). deriative of logL with respect of =(n/)-(summation over i=1 to n) and second derivative of logL with respect of   =-(n/ ^(2)). I( )=-expectation of[second derivative of logL with respect of   )= (n/ ^(2)). (B): Cramer-Rao lower bound of any unbiased estimator of   is [gamma\'( )]/l( )=1/(n/( ^(2)))=( ^(2))/n, expectation(mean)=(1/n)*(summation over i=1 to n(expectation(Xi)))=(1/n)*(n/ )=1/ . Therefore Xbar(mean) is an unbiased estimator of   . (C):derivative of logL with respect of   =0 then 1/ =Xbar(mean) and 1/(V( ))=expectation (-second derivative of logL with respect of ) then V( )=( ^(2))/n . Therefore   is an asymptotically N[(1/ ),( ^(2)/n)]