The Maximal Variance of Unilaterally Truncated Gaussian and Chi Distributions

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, $f(x;μ,σ)$, 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 $μ$. 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 $M^2(π-2)/2$ 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.

Figures (17)

• Figure 1: Ordered pairs $(\mu,\sigma)$ resulting in a fixed mean. For each value of $\mu$, the plot indicates the value of $\sigma$ resulting in a mean of 1 for a Gaussian distribution over the interval $x \in [0,\infty)$. Also shown is the resulting variance over the same interval, which appears to approach 1 as $\mu$ becomes increasingly negative. The calculations involved successively finer 2-D grid searches of $(\mu,\sigma)$ space.
• Figure 2: Gaussian probability density functions. Panel (a): three Gaussian probability density functions (pdfs) with support over the shaded interval, $x \in [0,\infty)$. For all three density functions, $\sigma=2$, but $\mu$ varies, as shown. The means over the shaded interval are indicated by the "$M$"s. They are approximately 1.0502, 1.5958, and 2.5752. The variances are 0.7964, 1.4535, and 2.5187, respectively. Panel (b): family of Gaussian pdfs with support over $x \in [0,\infty)$, fixed mean ($M(x)_{|x \ge 0}=1$) and varying $r$ values, which are indicated in red. The corresponding $\mu$ and $\sigma$ values are listed in Table \ref{['tab:mu_sig_from_r']}. The variance decreases with higher $r$ values, while the center of the pdfs, $\mu$, approaches the mean from the left.
• Figure 3: Dependence of parameters and variance on the $r$ value for Gaussian distributions over the interval $x \in [0,\infty)$, in the case of $M=1$. Panel (a): the $\mu$ and $\sigma$ parameters plotted as a function of the r. Panel (b): the variance plotted as a function of $r$.
• Figure 4: Plots of the variance vs. boundary height and higher central moments vs. the $r$ value. Panel (a): the variance, $\mathrm{Var}(x)_{|x \ge 0}$, plotted as a function of the relative height, $H$, at the truncation boundary for a Gaussian with fixed mean, $M(x)_{|x \ge 0}=1$, from Eq. (\ref{['var_M_H']}). Panel (b): higher central moments (CMs) of UTGDs. The 3rd (red) and 4th (black) CMs are plotted as a function of $r$. The solid lines indicate the standardized CMs (conventional skewness and kurtosis), and the dotted lines the unstandardized or unnormalized CMs assuming $M=1, a=0$. Note K has a minimum below the canonical value of 3 at $r \approx 1.87$. K--kurtosis; Kun--unnormalized kurtosis; S--skewness; Sun--unnormalized skewness.
• Figure 5: Plots of parameterization results for UTGDs obtained with approximating functions. Panel (a): Plots of the variance, $\mathrm{Var}(x)_{|x \ge 0}$, and $\sigma$ as a function of the $\mu$ parameter for Gaussian distributions having a fixed mean (of 1) over the interval $x \in [0,\infty)$, using approximating function 1 (given by Eq. (\ref{['sigma_exp_approx']})). The red dots represent the exact values (calculated with Eqs (\ref{['sigma_M_r']}) and (\ref{['var_M_r']})), and the black curves the functional approximation. Panel (b): Plot of the errors in the data shown in Panel (a). The black plots (A and C) represent errors in $\mathrm{Var}(x)_{|x \ge 0}$, while the red plots (B and D) represent errors in $\sigma$. Curves A and B are the differences, calculated as (approximate - exact). Curves C and D are the fractional differences, calculated as ((approximate-exact)/exact). Note the change in scale relative to the prior figure. Panel (c): Results for approximating function 2, given by Eq. (\ref{['approx_Lambert']}), for Gaussian distributions with a mean of 1 ($M=1$) over $x \in [0,\infty)$. Panel (d): Errors in the results shown in Panel (c)--i.e., approximations obtained with Eq. (\ref{['approx_Lambert']}). The plots in Panels (c) and (d) are otherwise as described above.

Theorems & Definitions (14)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• proof
• Theorem 2.7
• proof
• proof