---
title: Scaled Chi Distributions (UTSCDs)
url: https://www.emergentmind.com/topics/scaled-chi-distributions-utscds
type: topic
---

# Scaled Chi Distributions (UTSCDs)

Scaled chi distributions (abbreviated *UTSCDs*, Unilaterally Truncated Scaled Chi Distributions, in the literature) arise as the marginal distribution of the Euclidean norm $R = \sqrt{x_1^2 + \cdots + x_n^2}$ of a centered $n$-dimensional Gaussian vector $(x_1, \ldots, x_n) \sim N(0, \sigma^2 \mathbf{I}_n)$. When attention is restricted to the half-line $R \geq a$ for a fixed cutoff $a \geq 0$, the resulting truncated distributions play a central role in statistical modeling, goodness-of-fit calibration, and extreme-value theory. Recent research has determined tight bounds on their variance when a mean constraint is imposed, illuminating the structure of truncated moment spaces and offering precise methods for distributional fitting and quantile computation [2511.11566], [2404.05062].

## 1. Definition and Fundamental Properties

The $n$-dimensional radially symmetric Gaussian, $x \sim N(0, \sigma^2 I_n)$, induces a marginal distribution on $R = \|x\|$, known as a scaled chi distribution. Its probability density function is
\[
f_{R}(r) = \frac{2}{(\sqrt{2}\,\sigma)^n\,\Gamma(\tfrac n2)}\,r^{n-1}\,\exp\left(-\frac{r^2}{2\sigma^2}\right), \qquad r \ge 0.
\]
Here $n$ is the degrees of freedom (dimension), $\sigma > 0$ the scale, and $\Gamma(s)$ the Euler gamma function.

Unilateral truncation restricts attention to the domain $r \ge a$ for fixed $a \ge 0$, yielding the normalized truncated density
\[
f_{R|R \geq a}(r) = \frac{f_R(r)}{\displaystyle\int_a^\infty f_R(s)\,ds}
= \frac{2\,r^{n-1}\exp(-r^2/2\sigma^2)}{(\sqrt{2}\sigma)^n\,\Gamma(\frac n2, \frac{a^2}{2\sigma^2})}, \qquad r \ge a,
\]
where $\Gamma(s, x) = \int_x^\infty t^{s-1}e^{-t}\,dt$ denotes the upper incomplete gamma function.

This generalizes the central chi and half-normal distributions, with the scaled chi as a special case of a generalized chi-square quadratic form [2404.05062]. The mass beyond $a$ is $P\{R \ge a\} = \Gamma(\frac n2, \frac{a^2}{2\sigma^2})/\Gamma(\frac n2)$.

## 2. Calculation of Truncated Moments

Moments of the truncated distribution quantify its central tendency and dispersion subject to the truncation:
\[
M_k(a, n, \sigma) = E[R^k \mid R \geq a] = \frac{\int_a^\infty r^k f_R(r)\,dr}{\int_a^\infty f_R(r)\,dr}
= (\sqrt{2}\,\sigma)^k\,\frac{\Gamma(\frac{n+k}{2}, \frac{a^2}{2\sigma^2})}{\Gamma(\frac n2, \frac{a^2}{2\sigma^2})}.
\]
Key quantities include the truncated mean,
\[
M_1(a,n,\sigma) = \sqrt{2}\,\sigma\, \frac{\Gamma(\frac{n+1}{2}, \frac{a^2}{2\sigma^2})}{\Gamma(\frac n2, \frac{a^2}{2\sigma^2})},
\]
and the truncated second moment,
\[
M_2(a,n,\sigma) = 2\,\sigma^2\, \frac{\Gamma(\frac{n+2}{2}, \frac{a^2}{2\sigma^2})}{\Gamma(\frac n2, \frac{a^2}{2\sigma^2})}.
\]
For $k=2$, the truncated variance is $\operatorname{Var}(R \mid R \geq a) = M_2(a,n,\sigma) - [M_1(a,n,\sigma)]^2$. These formulas enable explicit computation using incomplete gamma special functions, with direct SciPy and R implementations for practical applications [2404.05062].

## 3. Maximal Variance under Fixed Mean and Cutoff

For applications requiring calibration to a fixed (finite) mean $M$ on $r \geq a$, one may optimize the variance over permissible scale $\sigma$ and dimension $n$. The supremum is analytically attained for $a = 0, n = 1$ (the half-normal case), with the maximal variance
\[
\sup\{\operatorname{Var}(R \mid R \geq a)\} = \frac{\pi - 2}{2}\,M^2,
\]
where $M = \sigma \sqrt{2/\pi}$ and hence $\sigma = M \sqrt{\pi/2}$.

No other choice of $n$ or $a$ achieves a larger variance under the truncated mean constraint. If $a$ is strictly positive, the maximum becomes a function of $a$ and the optimizing dimension $n_{\mathrm{vmx}}$—see next section for this dependency [2511.11566].

## 4. Dependency of Variance on Dimension and Cutoff

Fixing $M$ and $a$ yields an optimization problem over $n$. The truncated variance $\operatorname{Var}(R \mid R \geq a)$, viewed as a function of continuous $n \in \mathbb{R}$, exhibits a unique global maximum at $n_{\mathrm{vmx}}$, which increases monotonically with cutoff $a$ (formally with $r^2 = (a/\sigma)^2$). As $a \to 0$, $n_{\mathrm{vmx}} \to -1$, and the variance diverges, enabling arbitrarily high dispersion for vanishing cutoff when $n$ is not restricted to integer values.

Restricting $n$ to $\mathbb{Z}^+$ constrains the maximizer to integer values, creating a staircase in $n_{\mathrm{vmx}}^{\ddagger}(a)$ as $a$ increases. This suggests that optimal calibration for truncated variance depends sensitively on the dimensionality, with fractional $n$ outside traditional probabilistic interpretation but useful for extremal studies.

## 5. Computational Methods for PDF, CDF, and Quantiles

For practical evaluation, the scaled chi distribution admits closed-form expressions for PDF, CDF, and quantiles in terms of incomplete gamma functions:
- PDF: $f_Y(y) = \frac{1}{2^{k/2-1}a^{k}\Gamma(k/2)}\,y^{k-1}e^{-y^2/(2a^2)}$ for $y \geq 0$,
- CDF: $F_Y(y) = \gamma(k/2, y^2/(2a^2))/\Gamma(k/2)$,
- Quantiles: $F_Y^{-1}(p) = a\sqrt{2\,G^{-1}(k/2, p)}$, where $G^{-1}(\alpha, p)$ solves $\gamma(\alpha, G^{-1})/\Gamma(\alpha) = p$.

Python, R, and Matlab example codes for these calculations were provided by Das (2024), leveraging SciPy's `gammainc`, `gammaincinv` and analogous functions, as well as an open-source Matlab toolbox (github.com/abhranildas/gx2) supporting multiple exact and approximate computational methods (series-expansion "Ruben", characteristic function inversion "Imhof", inverse FFT, and ray-tracing, with tailored precision and speed for diverse use cases) [2404.05062].

| Software     | Function names                     | Methods Supported                            |
|--------------|-----------------------------------|----------------------------------------------|
| Python/SciPy | pdf_scaled_chi, cdf_scaled_chi     | Closed-form via incomplete gamma             |
| R (stats)    | pdf_scaled_chi, cdf_scaled_chi     | Closed-form via pchisq, dchisq               |
| Matlab (Das) | gx2cdf, gx2pdf, gx2inv             | ‘ruben’, ‘imhof’, ‘ifft’, ‘ray’              |

## 6. Calibration, Applications, and Theoretical Implications

The closed-form and optimization properties of UTSCDs position them as canonical models for statistical calibration under truncation, goodness-of-fit procedures, tail risk analysis, and simulation of radial norms. Model calibration can proceed by solving for $\sigma$ in $M = M_1(a, n, \sigma)$ for given $M, a, n$, then calculating desired moments and quantiles. The variance bounds facilitate testing and model fitting when only truncated means are observable.

A plausible implication is that the blow-up of variance for fractional $n \to -1$ in the $a \to 0$ limit provides theoretical upper bounds for dispersion under severe truncation, relevant for extremal value theory and robust statistics. The connection to generalized chi-square and quadratic forms opens avenues for further generalization, e.g., for noncentral or weighted sums.

## 7. Relation to Generalized Chi-Square and Quadratic Forms

Scaled chi distributions constitute a subclass of the generalized chi-square family, i.e., distributions of quadratic forms in Gaussian vectors. The generalized case, expressible as $Q(x) = x'Q_2x + q_1'x + q_0$, encompasses weighted sums of (possibly noncentral) chi-square variables and normal terms. Exact computation is enabled by Imhof-type characteristic function inversion and Ruben-type series expansion; Pearson moment-matching and saddlepoint/Fourier methods provide rapid approximations with quantified error properties.

Direct specialization to $Y = a\sqrt{X}$, $X \sim \chi^2_k$, retrieves the UTSCD structure, justifying use of incomplete gamma functions and the computational recipes previously described [2404.05062]. The relationship clarifies why scaled chi statistics are so broadly applicable, and situates the maximal variance result as a central component in the toolbox of extremal analysis and statistical calibration for truncated domains.

Source: https://www.emergentmind.com/topics/scaled-chi-distributions-utscds