Differentiability of Densities

Abstract

Suppose that $p_{\theta}$ is a probability density of sample $X$, $T$ is a mapping, $\textit{g}\kern0.5pt_{\theta}(t)$ is an induced probability density by $T$ and $k_{\theta}(x)$ is a conditional density given $T=t$. Then, the following results are proved under some conditions. (a) $L^2$-differentiability of the family $(\sqrt{p_{\theta}})$ is equivalent to that of $(\sqrt{\textit{g}\kern0.5pt_{\theta}})$ and $(\sqrt{k_{\theta}})$. (b) Regularity of the family $(p_{\theta})$ is equivalent to that of $(\textit{g}\kern0.5pt_{\theta})$ and $(k_{\theta})$.