Let X1, X2, ... Xn denote a random sample of size n from a continuous distribution with pdf
f(x,theta) = theta x^(theta-1) for 0 < x < 1 and theta > 0.
Show that the geometric mean = (X1X2X3...Xn)^(1/n), of the sample is a sufficient statistic for theta.
Find the maximum likelihood estimator of theta, and observe that it is a function of this geometric mean.