Guessing at pivots

Question

I'm trying to get a better idea of the intuition behind finding pivotal quantities. In the Casella & Berger statistical inference text, we have a $beta(\theta,1)$ pdf, $f_X(x)=\theta x^{\theta-1}$, $0 < x < 1$. The text says that $X^\theta$ is a good guess for a pivot, and then proceeds to verify that it is a pivotal quantity. My question is, what is the intuition that would lead one to believe that $X^\theta$ is a good guess?

Answer 1

In that case, just by looking at the density we can see the cdf (within the region where the density is nonzero) is $x^\theta$.

i.e. we know that $P(X\leq x) = x^\theta$ for $0<x<1$

But we also know that for continuous r.v.s $F_X(X)\,$* is standard uniform.

* (here suppressing the dependence on parameters, but they're still there of course)

So in the cases where we can easily write our pivot $Q=t(X;\theta)=F_X(X)$ we will automatically have that the transformed variable will have a distribution that doesn't depend on $\theta$ (since it's just $U(0,1)$).

Answer 2

Because the transformation theorem.

Equivalent to the answer above, but less pretty.

Define $Y = X^{\theta}$. Then $X = Y^{1/\theta}$, then $$ f_Y(y) = \frac{\theta}{\theta} [y^{1/\theta}]^{\theta-1}y^{1/\theta - 1} = 1. $$

It's free of $\theta$.