Suppose X1 X2 Xn is a random sample from the uniform distrib

Suppose X1, X2,..., Xn is a random sample from the uniform distribution over the interval (0, theta). A natural estimator of theta is the largest observation X(fl). It can be shown that a) Find a multiple of X that is an unbiased estimator of theta, and find its MSE.| b) Find a multiple of X(n) that is an unbiased estimator of theta, and find its MSE. C) Find the relative efficiency of the estimator in part (a) compared to the efficiency of the estimator in part (b).

Solution

As all xis follow uniform distribution

a) Multiple of X which is an unbiased estimator for theta = theta/n

AS each xi has prob 1/theta, x1+x2+....Xn = n /theta

Hence estimator for theta = theta/n, where n is the no of xis.

b) Mean square error =

= n(theta)^2+ theta^2(1+1/2^2+...+1/n^2) -sum of 2theta^2/i^2

----------------------

c) Efficiency = nearly 100%