point estimate question: sufficiency, bias, order stat
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.
(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?