Problem 3 Let X1 X2 X3 X be a random sample from the follo

Problem 3 Let X1, X2, X3.. . ., X, be a random sample from the following distribution fx(x) = where theta belongs to [-2, 2] is an unknown parameter. We define the estimator theta n as thetan = 12X-6 to estimate theta. a. Is thetan an unbiased estimator of theta? b. Is thetan a consistent estimator of theta? c. Find the mean squared error (MSE) of psi n.

Solution

Given that tn = 12 xbar -6

To check whether unbiased estimator

find E(tn) = E(12 x bar-6) = 12E(xbar)-6

E(xbar) = integral of x f(x) dx = 12(7t/12) -6 = 7t-6

Thus Mean of this is equal to population parameter , hence unbiased.

b) consistent as it is unbiased

c) MSE of tn = E(X^2) = 5t/12