1*. Suppose that Xi....,Xn is a random sample of size n on X, where X has pdf f(x, = -₉₂ X³e^-2218, X> 0, and 0 is a positive parameter.

(a)    Let Y = 2X²/0. Show that Y xi. Using this (or otherwise), give the mean and variance of X².

(b)   Let U = ;  X. State the distribution of U, and give its mean and variance.

(c)    Write down the likelihood function (for 0) based on the sample of size n, and identify a sufficient statistic.

(d)   Find E(X). Using this, give a "method of moments" estimator (MME) e of O. Give its mean and variance, and show that it is consistent. Is it a sufficient statistic?

(e)    Find the MLE B. Give its mean and variance and show that it is consistent. Is it a sufficient statistic?

(f)    Is the MLE 0 an efficient estimator of 0?

(g)  State a result about the asymptotic distribution of B. Give an approximate 100(1-a) confidence interval for 9 based on this large•sample distribution.

(h)   Consider the parameter tp ty(0) = P(X > 2), Express _iG in terms of O.

(i)    Give the MLE of ti). Is it a consistent estimator of IP?

(j)    Consider testing Ho : 9 = 00 against Hi : 9 = 01 at level a (where < 00). Show that the most powerful (MP) test rejects Ho in favour of H1 if t < c, where t is the observed value of the statistic T = EL, V. Explain how you would find the constant c.

(k)   Is the MP test in (j) a UMP test of Ho against HA : 0 < 9o. Explain your answer.

(I) Describe, as completely as possible, the likelihood ratio test (LRT) with significance level a of Ho : 9 = e_o vs the two-sided alternative HI : 0 0 00.