Exact mean and covariance formulas after diagonal transformations of a multivariate normal

TL;DR

This paper derives exact formulas for the mean and covariance of a multivariate normal vector after applying diagonal transformations to each component, providing theoretical and numerical validation.

Contribution

It introduces two methods for calculating the mean and covariance after diagonal transformations of a multivariate normal, including series expansion and transform techniques.

Findings

Exact formulas for mean and covariance after diagonal transformations.

Comparison of theoretical results with numerical simulations.

Methods applicable to various functions $f_i$ for practical estimation.

Abstract

Consider $\boldsymbol X \sim \mathcal{N}(\boldsymbol 0, \boldsymbol \Sigma)$ and $\boldsymbol Y = (f_1(X_1), f_2(X_2),\dots, f_d(X_d))$. We call this a diagonal transformation of a multivariate normal. In this paper we compute exactly the mean vector and covariance matrix of the random vector $\boldsymbol Y.$ This is done two different ways: One approach uses a series expansion for the function $f_i$ and the other a transform method. We compute several examples, show how the covariance entries can be estimated, and compare the theoretical results with numerical ones.