Questions relating to UMVUE/transformations

[abeliando]

So one of my assignments was to find a UMVUE of a Rayleigh distribution, that is the pdf
f(x;θ)=2θ^-1x*exp(-x²/θ).
the suggestion was to use U=Ʃx²_i. When I do this, I do a transformation with U_i=X²_i and find that U_i has an exponential distribution, and the U=ƩU_i follows a gamma(n,θ), so then n^-1E(U)=θ, so then U-bar is the UMVUE of θ. I think this is correct.

So then I am asked to see if it attains the CRLB, and I use the fact that U is Gam(n,θ) to show that yes indeed, it does reach the CRLB. All is well and good. Also, I'm asked to find the estimators of θ² and θ^-1, and find their variances. I did this, and it all relied on the fact that it was gamma distributed.

On the next problem, I'm given that X follows a Weibull, that is, has the pdf
f(x;θ)=α^-1βx^β-1*exp(-x^β/α), where β (>0) is known, and it suggests starting off with U=ƩX^β_i. So when I make this transformation, I find the same thing, that U_i is exponential, so U=ƩX^β_i is distributed Gam(n,α).

So here's why I'm confused. In both cases, I started with a pdf, transformed it to a different variable, but otherwise, finding the estimators and their variances aren't based on their original distribution, but the transformed distribution, which is the same. So I'd imagine that I just cut and paste my original work. It can't be that easy, right? What am I missing? The problems in the text are assigned right after each other, so... I'm assuming I'm wrong. Any help?

 

[Ray Vickson]

What's a UMVUE? What's CRLB? (I *think* I know what the last one means, but how can I be sure?)

 

[abeliando]

UMVUE = Uniformly Minimum Variance Unbiased Estimator, CRLB = Cramer Rao Lower Bound (defined as the first derivative of a function of a parameter squared, over the information contained by the set of data) (τ(θ)')²/I_X

So by Lehmann-Scheffe theorems, if I take a complete sufficient statistic that is unbiased for an parameter, that statistic is the UMVUE. In my cases, the suggested statistics that I started off with are complete sufficient, and then taking the average is still complete sufficient.

CRLB is the function that states what the theoretical minimum variance for an estimator is, and then I just take the variance to see if it actually works

 

[abeliando]

Hm, this is a little bit tricky. If I did my work correct, I suppose it does make sense, and rely on the previous distribution. I am confused about the two U's (U_1=ƩX²_i and U_2=ƩX^β_i apparently having the same distribution. And I guess what I' m saying is, is that in the first case, if X follows a Rayleigh distribution, then U_1 follows a Gamma(n,θ) distribution. Also if X follows a Weibull distribution, then U_2 follows a Gamma(n,θ) distribution. So it does depend on the original distribution. Is this correct?

This makes sense. It would also make sense that the work would be same, I guess.

Oh, and the information contained in the set will (should) be different (I haven't evaluated it yet).

I think I figured it out, yes?