Hello, I'm look for some help on how to solve this problem. Let f(x; \theta ) = g( \theta )h(x), a(\theta) \le x \le b(\theta), \theta \in \Omega ,

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

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

a) Show that S = max ( is a sufficient statistic for 

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

 = 

c) Show that P(S;) =  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 ( ) and explain how you will determine the constants