7.5.3 Let X1, X2,..,Xn denote a random sample of size n from a distribution with pdf f(x; è) =èx^(è-1), 0<x<1, 0 elsewhere, and è>0.
Show that the geometric mean (X1X2...Xn)^(1/n) of the sample is a complete sufficeint statistic for è.
Find the maximum likelihood estimator and observe that it is a function of this geometric mean.
This solution provides a proof to show that the geometric mean of the sample is a complete sufficient statistic for e. It also finds the maximum likelihood estimator and also shows how it is a function of the geometric mean. All steps are shown with brief explanations.