Let's say I have an unbiased estimator $u(\underline x)$ for function $v(\theta)$ where $\theta$ is a parameter of the distribution of $x$, and $T(\underline x)$ which is a sufficient statistic for $\theta$. To use Rao-Blackwell to diminish the variance of the estimator, I need to find $E(u(\underline x)|T(\underline x))$.

For this I need to find the joint PDF $f_{u,T}(u(\underline x),T(\underline x))$ and marginal PDF $f_{T}(T(\underline x))$ so I can calculate ${f_{u,T}(u(\underline x),T(\underline x))}\over{f_{T}(T(\underline x))}$

What is the recommended fail-safe strategy for finding the joint and marginal PDFs if I'm not lucky enough to have a distribution where those PDFs are "obvious" by inspection or from prior knowledge?

Often $T(\underline x)$ will be an order statistic or some function of $\underline x$ that happens to tractable closed-form expression for its density, but even in those cases I'm stumped on how to munge that back into the original PDF for $\underline x$ in order to get the joint PDF of the data and the sufficient statistic.

  • $\begingroup$ If this question is still active, it is usually easier to figure out first the distribution of $X$ conditional on $T(X)$, rather than the joint of $u(X)$ and $T(X)$. $\endgroup$
    – Xi'an
    Commented Nov 1, 2015 at 15:18


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