Sample distribution theory using Coarea Formula

Abstract

Let $\left(\Omega ,\Sigma ,p\right)$ be a probability measure space and let $X:\Omega \to {\mathbb{R}}^k$ be a (vector valued) random variable. We suppose that the probability $p_X$ induced by $X$ is absolutely continuous with respect to the Lebesgue measure on ${\mathbb{R}}^k$ and set $f_X$ as its density function. Let $\varphi :{\mathbb{R}}^k\to {\mathbb{R}}^n$ be a $C^1$-map and let us consider the new random variable $Y=\varphi (X):\Omega \to {\mathbb{R}}^n$. Setting $m:=\max\{\mbox{rank }(Jϕ(x)):x\in{\mathbb{R}}^k\}$, we prove that the probability $p_Y$ induced by $Y$ has a density function $f_Y$ with respect to the Hausdorff measure ${\mathcal{H}}^m$ on $\varphi ({\mathbb{R}}^k)$ which satisfies \begin{align*} f_Y(y)= \int_{ϕ^{-1}(y)}f_X(x)\frac{1}{J_mϕ(x)}\,d{\mathcal{H}}^{k-m}(x), &\quad \text{for ${\mathcal{H}}^m$-a.e.}\quad y\inϕ({\mathbb{R}}^k). \end{align*} Here $J_m\varphi$ is the $m$-dimensional Jacobian of $\varphi$. When $J\varphi$ has maximum rank we allow the map $\varphi$ to be only locally Lipschitz. We also consider the case of $X$ having probability concentrated on some $m$-dimensional sub-manifold $E\subseteq {\mathbb{R}}^k$ and provide, besides, several examples including algebra of random variables, order statistics, degenerate normal distributions, Chi-squared and "Student's t" distributions.

arXiv comment22 pages