Weighted Chi-Square Distribution

Updated 20 August 2025

Weighted chi-square distribution is a linear combination of independent chi-square or squared Gaussian variables with nonuniform, nonnegative coefficients.
Analytical methods—such as moment generating functions, saddle-point approximations, and Laplace inversions—provide explicit density and distribution expressions.
These distributions underpin applications in multivariate analysis, goodness-of-fit tests, random matrix theory, and high-dimensional statistics with precise asymptotic and error insights.

A weighted chi-square distribution refers to the law of a sum or process in which independent (central or noncentral) chi-square random variables or squared Gaussian processes enter with non-uniform, nonnegative (possibly real or complex) coefficients. This class includes distributions arising from quadratic forms, spectral decompositions of Wishart matrices, sums of independent but heterogeneously weighted squared normals, and the marginal distributions of statistics such as the Pearson chi-square, Cramér–von Mises, and Anderson–Darling statistics. Weighted chi-square distributions appear naturally in multivariate analysis, random matrix theory, the theory of quadratic forms, goodness-of-fit testing, and the asymptotics of stochastic processes. Their properties are determined by both the weights and the structure of the underlying variables.

1. Analytical Representations: Density and Distribution Functions

The fundamental object is a linear combination

$Y = \sum_{j=1}^n \lambda_j \chi^2_{k_j}(\delta_j)$

where $\lambda_j \neq 0$ are real weights, and $\chi^2_{k_j}(\delta_j)$ are (possibly noncentral) chi-square variables with degrees of freedom $k_j$ and noncentrality parameter $\delta_j$ . The distribution function and density can be derived by several methods:

Moment generating function (MGF) techniques: For central independent components, the MGF factorizes as $M(t)=\prod_j (1-2\lambda_j t)^{-k_j/2}$ , enabling inversion using partial fractions, residue calculus, or Laplace inversion (Unsal et al., 2021).
Explicit closed-form expansions: For example, when $n=2$ or $3$, the PDF is a sum over terms of the form $C e^{-x/(2\lambda_j)} x^{r}$ with coefficients determined by the partial fraction expansion (Unsal et al., 2021). The corresponding CDFs are linear combinations of incomplete gamma functions in the transformed argument.
Saddle-point and large-N asymptotics: In the case of highly degenerate or structurally repeated weights, as in trigonometric spectra, the density is determined by a residue expansion for all $N$ , but a saddle-point integral approximation controls the large deviation behavior and captures non-Gaussian tails (Egorov et al., 23 Dec 2024).
Integral representations for multivariate and correlated settings: For joint or marginal distributions arising from general quadratic forms or from Wishart matrices, the Laplace transform has the universal form

$\mathcal{L}(t_1,\ldots,t_p) = |I_p + 2 R T|^{-v/2} \exp\left\{-\mathrm{tr}[T (I_p + 2RT)^{-1}A]\right\},$

with $R$ the correlation matrix and $A$ the noncentrality (Royen, 2016). For $p=2,3$ , (p-1)-dimensional integral inversions are computationally feasible.

Case	Density Expression	Method
$n=2$	Explicit sum of exponentials and powers	Partial fractions (Unsal et al., 2021)
Large $n$ (degenerate weights)	Saddle-point integral, Gaussian core, non-Gaussian tails	Residue, saddle point (Egorov et al., 23 Dec 2024)
Multivariate	(p-1)-variate integral over explicit kernel	Laplace inversion (Royen, 2016)

2. Mixture Representations, Stein Equations, and Characterizations

Several important theoretical frameworks characterize or approximate weighted chi-square distributions:

Mixtures of scaled chi-square laws: Noncentral chi-square distributions with integer weights can be represented as Poisson mixtures of central chi-square distributions with varying degrees of freedom. In matrix statistical settings (e.g., the scalar Schur complement of a Wishart matrix), additional randomization over a beta or noncentral beta law leads to mixture coefficients given by explicit integrals involving beta and Poisson weights (Siriteanu et al., 2015). This structure enables explicit computation of moments and tail probabilities.
Stein's method characterization: Weighted sums of independent chi-square variables admit a Stein-type differential equation

$\mathcal{T} f(x) = \sum_{k=0}^r (-2)^k (\mu_k + \Lambda_k x) f^{(k)}(x)$

so that $\mathbb{E}\mathcal{T} f(Y) = 0$ characterizes the law of $Y$ (Chen et al., 2020). Here, $\Lambda_k$ and $\mu_k$ are combinatorial sums over products of weights and degrees of freedom, and $f^{(k)}$ is the $k$ th derivative. This operator enables quantitative bounds for distributional approximation (e.g., normal or chi-square limits in goodness-of-fit testing).

3. Asymptotic and Extremal Properties

Weighted chi-square distributions often arise in the limit theory of test statistics and stochastic processes:

Tail asymptotics for supremum functionals: For stochastic processes of the form $\chi^2_{(b)}(t) = \sum_{i=1}^n b_i^2 X_i^2(t)$ indexed by $t\in T$ , with $w(t)$ a deterministic weight, the tail behavior of the normalized sup-statistic

$S(u) = \sup_{t\in T} \frac{\chi^2_{(b)}(t)}{w^2(t)}$

admits precise asymptotic expansion as $u\to\infty$ , governed by the minimal value and regularity of $w(t)$ , the local covariance structure of $X(t)$ , and constants such as Pickands or Piterbarg constants (Ji et al., 2018). Boundedness of $S(u)$ is equivalent to finiteness of an exponential integrability condition on $w(t)$ .

Density bounds: Dimension-free two-sided bounds for the maximum value of the PDF of a weighted chi-square sum are given in terms of explicit functions of the weight vector $\lambda$ , e.g.

$c_0 (A_1A_2)^{-1/4} \leq M(W_0) \leq c_1 (A_1A_2)^{-1/4}$

with $A_1 = \sum \lambda_k^2$ , $A_2 = \sum_{k=2}^n \lambda_k^2$ (Bobkov et al., 2020). This result facilitates control of anti-concentration and moderate deviation probabilities in high-dimensional and infinite-dimensional settings.

Central limit and large deviation regimes: For large ensembles with symmetric or periodic weights (e.g., trigonometric spectra), the central region exhibits Gaussian behavior (variance dictated by the sum of squared weights), but the tails remain non-Gaussian and reflect the precise structure of the weights (Egorov et al., 23 Dec 2024).

4. Goodness-of-Fit Testing: Weighted Histograms, Pearson, and Runs-Based Statistics

Weighted chi-square statistics form the basis for numerous goodness-of-fit and homogeneity tests in settings with weighted observations:

Weighted histogram tests: Classical Pearson-type tests are generalized to accommodate the variance-covariance structure induced by random weights; test statistics are constructed from the generalized inverse of the covariance (excluding the bin with the least information, as indicated by $p_{i0}/\hat{r}_i$ or similar ratios), leading to asymptotically valid chi-square tests whose size (type I error rate) is close to nominal (Gagunashvili, 2013, Gagunashvili, 2016). For Poisson-weighted and unnormalized cases, modified statistics with degrees of freedom adjusted for normalization are introduced.
Runs-based statistics: Sensitivity to local (ordered) deviations is enhanced by partitioning the data sequence into weighted runs—the maximal sum of squared normalized deviations within a run forms the test statistic $T$ (Beaujean et al., 2010). Exact calculation for the distribution of $T$ uses integer partition combinatorics, and efficient Monte Carlo algorithms are proposed for large $N$ . Power studies demonstrate significant improvements over classical unweighted chi-square tests in identifying local anomalies.
Non-asymptotic error bounds and confidence sets: The finite-sample distribution of Pearson's chi-square and other statistics is shown to be well approximated by their Gaussian counterparts, with explicit total variation and quantile coupling error rates (Bax et al., 2023). This allows construction of confidence intervals for functionals of the probability weights (e.g., negative entropy) and justifies Gaussian and chi-square critical value usage in moderate $n$ .

5. Connections to Multivariate and Random Matrix Theory

Weighted chi-square distributions underlie several results in multivariate analysis—specifically, the spectral behavior and inferential procedures for covariance matrices:

Singular Wishart matrices and eigenvalue distributions: For Wishart matrices with singular population covariance, the joint density of sample covariance eigenvalues (scaled) is asymptotically approximated by independent chi-squared laws with varying degrees of freedom, conditioned on "spiked" population eigenvalues (Shimizu et al., 2023). This fact underpins the use of $F$ - and chi-square-based ratio statistics for hypothesis testing on individual population eigenvalues in high-dimensional contexts.
Quadratic forms and diagonalizations: Quadratic forms in Gaussian vectors, under appropriate diagonalizing transformations, reduce to weighted sums of independent chi-square variates; explicit integral representations using the Fisher–Bingham distribution, holonomic gradient methods, and related ODE systems provide computationally efficient routes to cumulative probabilities and densities (Koyama et al., 2015). Applications include the calculation of probabilities for balls in multivariate normal space and the paper of specific statistics (e.g., Cramér–von Mises, Anderson–Darling) structured as infinite weighted chi-square sums.

6. Numerical and Algorithmic Techniques

Computation of densities, CDFs, and $p$ -values for weighted chi-square distributions requires specialized techniques. Key methodologies include:

Partial fraction and residue methods: Used for explicit inversion of the MGF in the finite-term case, yielding numerically stable sums of exponential and polynomial terms (Unsal et al., 2021).
Series and integral expansions: For moderate dimensions, multidimensional inversions of the Laplace transform yield (p-1)-dimensional explicit integrals.
Monte Carlo simulation: Necessary for statistics defined via maxima or complex combinatoric structures (e.g., weighted runs), Monte Carlo approaches scale efficiently to large $N$ and admit error quantification (Beaujean et al., 2010).
Holonomic gradient methods: For high-dimensional or noncentral cases, reducing the task to ODE integration of a holonomic system originally satisfied by the normalizing constant (e.g., Fisher–Bingham) ensures stability and high accuracy (Koyama et al., 2015).

7. Applications, Implications, and Limitations

Weighted chi-square distributions pervade a range of domains, including but not limited to:

Statistical inference: Power studies for weighted chi-square-based statistics reveal marked improvements in sensitivity for detecting local deviations and anomalies in data, especially in Monte Carlo and high energy physics applications (Beaujean et al., 2010, Gagunashvili, 2013, Gagunashvili, 2016). Robust tests accommodate full covariance structure induced by weighting and adjust for normalization uncertainty.
Random matrix and signal processing theory: Characterizations of Schur complement distributions, eigenvalue testing in spiked models, and SNR analysis rely on weighted chi-square or mixtures thereof (Siriteanu et al., 2015, Shimizu et al., 2023).
Functional and infinite-dimensional statistics: Nonasymptotic bounds for probabilities and densities in Hilbert or Banach spaces depend critically on explicit control of density peaks and anti-concentration inequalities for weighted chi-square sums (Bobkov et al., 2020). This is crucial for analysis in high-dimensional and functional data analysis.

However, limitations are present:

Complexity of exact expressions: As the number of terms increases or the weight structure becomes more intricate, explicit inversion (via residues or partial fractions) becomes impractical, requiring resort to asymptotics, saddle-point approximations, or simulation.
Strong assumptions in asymptotic regimes: Approximations for marginal eigenvalue distributions or tail asymptotics depend on "infinite dispersion," "spiked," or "locally stationary" assumptions (Shimizu et al., 2023, Ji et al., 2018), whose validity must be checked in the application at hand.
Non-Gaussianity outside the CLT regime: Even when the central part of the distribution is Gaussian, the tails may display markedly non-Gaussian decay due to the weight structure (Egorov et al., 23 Dec 2024).

In summary, the weighted chi-square distribution is a central object in probability, statistics, and their applications, combining combinatorial, analytical, and computational features. Its paper illuminates the behavior of finite and infinite-dimensional quadratic forms, provides powerful tools for model assessment and hypothesis testing in weighted or multivariate settings, and necessitates a nuanced approach integrating both tractable special cases and general high-dimensional theory.