The Maximal Variance of Unilaterally Truncated Gaussian and Chi Distributions

TL;DR

This paper investigates the maximum variance bounds of unilaterally truncated Gaussian and chi distributions, providing theoretical proofs, calibration methods, and numerical evidence for variance behavior across parameters.

Contribution

It establishes variance bounds for truncated Gaussian distributions and analyzes variance maxima in scaled chi distributions, introducing new calibration functions and insights.

Findings

Variance of truncated Gaussian is bounded by (M - a)^2.

Maximum variance of scaled chi distribution occurs at specific parameters, notably n=1 and a=0.

Variance can grow unbounded as cutoff approaches zero for certain degrees of freedom.

Abstract

This work explores the bounds of the variance of unilaterally truncated Gaussian distributions (UTGDs) and scaled chi distributions (UTSCDs) with fixed means. For any arbitrary Gaussian distribution function, $f(x;\mu,\sigma)$, with a fixed, finite mean $M$ on the truncated domain $x \ge a$, where $a \in \mathbb{R}$, it is proven that the variance is bounded: specifically, $\sup \mathrm{Var}(x)_{|x \ge a}= \sup \mathrm{Var}(x)_{|x \le a} =(M-a)^2$. For a fixed cutoff, $a$, the variance can be considered a function of only $M$, $a$, and the location parameter $\mu$. Examples of such approximating functions, which can be used for model calibration, are developed in addition to other, related calibration methods. For UTSCDs, numerical evidence is presented indicating that for $n \in \mathbb{Z+}$ degrees of freedom, or dimensions, and a fixed, finite mean, the variance, $\mathrm{Var}(R)$, over $R \in [a,\infty)$ reaches its maximum value $M^2(\pi-2)/2$ at $a=0$, $n=1$. For a fixed cutoff value, there is a local maximum in the variance as a function of $n$, and the number of dimensions resulting in the maximal variance, $n_{\mathrm{vmx}}$, increases with cutoff value. However, for $n \in \mathbb{R}$, as the cutoff approaches $0$, $n_{\mathrm{vmx}}$ approaches $-1$, while $\mathrm{Var}(R)$ appears to grow without bound.