Sample distribution theory using Coarea Formula

TL;DR

This paper extends the distribution theory for random variables transformed by smooth maps using the Coarea Formula, providing explicit density formulas with applications to various distributions and cases.

Contribution

It introduces a generalized density formula for transformed random variables using the Coarea Formula, accommodating cases with lower rank maps and manifolds.

Findings

Derived explicit density formulas for transformed variables.

Extended the theory to locally Lipschitz maps with maximum rank.

Provided examples including algebraic, order statistics, and classical distributions.

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 $\phi:{\mathbb{R}}^k\to {\mathbb{R}}^n$ be a $C^1$-map and let us consider the new random variable $Y=\phi(X):\Omega\to{\mathbb{R}}^n$. Setting $m:=\max\{\mbox{rank }(J\phi(x)):x\in{\mathbb{R}}^k\}$, 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 $\phi({\mathbb{R}}^k)$ which satisfies \begin{align*} f_Y(y)= \int_{\phi^{-1}(y)}f_X(x)\frac{1}{J_m\phi(x)}\,d{\mathcal{H}}^{k-m}(x), &\quad \text{for ${\mathcal{H}}^m$-a.e.}\quad y\in\phi({\mathbb{R}}^k). \end{align*} Here $J_m\phi$ is the $m$-dimensional Jacobian of $\phi$. When $J\phi$ has maximum rank we allow the map $\phi$ 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.