the largest order stat is suff for theta
I believe that
If that's right, then is unbiased for theta
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!
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?
the likelihood functions factors into the five parts
the only part involving the data that cannot be separated from theta is the one with the largest order stat.
that is always the case when dealing with a parameter greater than the random variables.
When the parameter is a lower bound, the the smallest order stat is suff for that parameter.
So, to continue to part (d), I think I need to check 'attaining the Cramer-Rao bound'.
Given the likelihood function for n observations above, ,
Now I think I should check the 'attaining Cramer-Rao lower bound' condition, whether or not the score function is linear with the estimator function
should I find b and g of theta here?
I am just copying the formula from the book. How do N! mathematicians do that?...
Another attempt at (d) By considering the condition that a function of is unbiased for , determine whether there is a better unbiased estimator for .
If is an unbiased etimator for based on a random sample , then
("I" here denotes information - what it the real Latex code for this letter? is it Greek?...)
Now I find the variance of my unbiased estimator and compare to the above variance threshold
then
I am concerned that the final Var does not depend on theta - theta squared cancelled out in the computation.
Now I understand one would see how this variance compares with
???