A statistic $A(X)$ is first order ancillary if $\mathbb{E}_{\theta}[A(X)]$ does not depend on $\theta$ where $X \sim P_{\theta}$. Show when distribution of $A(X)$ is independent of $\theta$, $\mathbb{E}_{\theta}[A(X)]$ is independent of $\theta$?
My try:
I think we need to show $\mathbb{E}_{\theta}[A(X)] = \int_{-\infty}^{+\infty}A(x)p_{\theta}(x)dx$ is independent of $\theta$ but how?
$\endgroup$1 Answer
$\begingroup$By the definition of an ancillary statistic, the function $p_\theta$ is the same for all $\theta$, therefore so is the integral.
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