estimation problem for two-parameter weibull distribution

Question

Suppose the two-parameter Weibull distribution is given by the pdf $$ f(x;a,b) = \left(\frac{x}{a}\right)^b\frac{b}{a}\exp\left\{-\left(\frac{x}{a}\right)^b\right\}, $$ where $x,a,b>0$. I am interested in estimating $a^b$ based on a sample of size $N$. Are there estimators with known (in closed form) distributions (for any finite $N$, not asymptotically)?

Answer 1

I don't know of a way to estimate it directly (nor do I know why you'd want to). But there are good methods for estimating the parameters individually, and then you can take whatever function of both of them you want.

http://interstat.statjournals.net/YEAR/2000/articles/0010001.pdf http://en.wikipedia.org/wiki/Weibull_distribution#Parameter_estimation http://contentdm.lib.byu.edu/cdm/ref/collection/ETD/id/2543 etc.

I don't think there are closed form solutions for the parameters directly. One can derive implicit equations for the parameters and solve.