## Tokyo Journal of Mathematics

### Differentiability of Densities

Yoichi MIYATA

#### 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 $(k_{\theta})$.

#### Article information

Source
Tokyo J. Math., Volume 25, Number 1 (2002), 153-163.

Dates
First available in Project Euclid: 5 June 2009

https://projecteuclid.org/euclid.tjm/1244208942

Digital Object Identifier
doi:10.3836/tjm/1244208942

Mathematical Reviews number (MathSciNet)
MR1908219

Zentralblatt MATH identifier
0996.62006

#### Citation

MIYATA, Yoichi. Differentiability of Densities. Tokyo J. Math. 25 (2002), no. 1, 153--163. doi:10.3836/tjm/1244208942. https://projecteuclid.org/euclid.tjm/1244208942

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