Scaled Chi Distributions (UTSCDs)

• Scaled Chi Distributions (UTSCDs) are the marginal distributions of the Euclidean norm of an n-dimensional Gaussian vector, modified via unilateral truncation at a fixed cutoff.
• Their computation utilizes closed-form expressions with incomplete gamma functions to derive PDFs, CDFs, and quantiles, enabling precise statistical calibration.
• They are crucial for applications in model fitting, goodness-of-fit testing, and extreme-value theory, especially under variance optimization constraints.

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 (Petrella, 14 Nov 2025, Das, 2024).

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 (Das, 2024). 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 (Das, 2024).

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 (Petrella, 14 Nov 2025).

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) (Das, 2024).

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 (Das, 2024). 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.