Hello, I'm look for some help on how to solve this problem.

Let f(x; ) = g( )h(x), a()xb(), , where a() decreases and b( ) increases with . Suppose g() is a differentiable function of and that we have a random sample X1,...,Xn from this pdf. Let X1:n<...<Xn:n be the corresponding order statistics.

a) Show that S = max (a1(X1:n,b1(Xn:n)) is a sufficient statistic for

b) Prove then that the Minimum Variance Unbiased Estimator of is

n = Sng(S)g(S)

c) Show that P(S;) = g(S)g() is a pivotal quantity and then derive its distribution

d) Use the pivotal quantity P(S;) above to obtain a 100(1- )% confidence interval for g() of the form (c1g(S),c2g(S) ) and explain how you will determine the constants c1andc2