point estimate question: sufficiency, bias, order stat

Question.

Suppose that is a random sample from a population with density for , where .

(a) is a sufficient statistic for ?

(b) Derive the density of the last order statistic .

(c) Find an unbiased estimator of that is a function of .

(d) By considering the condition that a function of is unbiased for , determine whether there is a better unbiased estimator for .

I've just started stat inference and my confidence is so low, I feel like threading on seashells... basically I worked through most of proofs but I don't see how I can apply them. This is a sample exam question (to which I don't have an answer). So I'd appreciate your pointers.

Answer.

(a) sufficiency of

Should I apply factorisation theorem here to factorise the density of **Y **into two parts: one is the function of and , another is a function of **y** only?

Another thought, given the fact that y is limited by , could it be that y somehow depends on and therefore the highest order statistic cannot be a sufficient statistic simply because of this dependency?

(b) Density for

find anti-derivative of

- for a single observation

(c) unbiased estimator of that is a function of

I cannot 'see' it just from looking at the distribution, so I was thinking to use the definition of the unbiased estimator, but the integration seems like a puzzle.

Define a function of . Then

according to the definition of an unbiased estimator.

Then I would try to find form the equation

If I do that, I get an integral of a product of and , on the left side, and an expression involving theta and n on the right side - is it solvable?

Any advice? Thanks!

expected value of i-th order statistic

One more (general) question related to (c), how do you find an expected value of an i-th order statistics - do you use the usual formula (summation or integration)

I can find the density of i-th order stats, but and what do I put as ? y? or I need to derive a formula for , based on the distribution given?