A note on sum and difference of correlated chi-squared variables

Alberto Ferrari

arxiv: 1906.09982 · v1 · pith:RYN4KOQSnew · submitted 2019-06-24 · 📊 stat.ME · math.ST· stat.TH

A note on sum and difference of correlated chi-squared variables

Alberto Ferrari This is my paper

Pith reviewed 2026-05-25 17:17 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH

keywords chi-squared distributiongamma distributionVariance-Gamma distributioncorrelated random variablesapproximate distributionssum and difference

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The pith

Sums and differences of linearly correlated chi-squared variables approximate gamma and Variance-Gamma distributions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives approximate distributions for the sum and difference of linearly correlated chi-squared random variables. It shows that the sum reduces to a gamma distribution and the difference to a Variance-Gamma distribution. These approximations are flexible and extend to multiple variables or to gamma variables with common parameters. A reader would care because they provide convenient parametrizations for analyzing such variables in statistical applications. The fits are tested on simulated data.

Core claim

Approximate distributions for sum and difference of linearly correlated χ² distributed random variables are derived. It is shown that they can be reduced to conveniently parametrized gamma and Variance-Gamma distributions, respectively. The proposed distributions are very flexible, and the one for sum in particular has straight-forward generalizations to cases where multiple χ² variables with different parameters are involved. The results promptly extend to every sum of gamma variables with common scale and to every difference between gamma variables with common shape and scale.

What carries the argument

The reduction of the sum to a gamma distribution and the difference to a Variance-Gamma distribution under linear correlation.

If this is right

The sum approximation generalizes to multiple chi-squared variables with different parameters.
The distributions extend to sums of gamma variables with common scale.
The distributions extend to differences of gamma variables with common shape and scale.
The approximations are useful for researchers working on gamma-distributed variables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These approximations might simplify calculations in applied statistics where chi-squared statistics appear from underlying correlations.
Accuracy could be checked by comparing higher moments beyond those used in the parametrization.
The approach might extend to other common scale families if the linear structure is preserved.

Load-bearing premise

The linear correlation structure between the chi-squared variables permits reduction to gamma and Variance-Gamma parametrizations without further constraints.

What would settle it

A Monte Carlo simulation of two correlated chi-squared variables where the empirical distribution of their sum deviates substantially from the fitted gamma distribution.

read the original abstract

Approximate distributions for sum and difference of linearly correlated $\chi^{2}$ distributed random variables are derived. It is shown that they can be reduced to conveniently parametrized gamma and Variance-Gamma distributions, respectively. The proposed distributions are very flexible, and the one for sum in particular has straight-forward generalizations to cases where multiple $\chi^{2}$ variables with different parameters are involved. The results promptly extend to every sum of gamma variables with common scale and to every difference between gamma variables with common shape and scale. The fit of the distributions is tested on simulated data with remarkable results.The approximations presented are expected to be especially useful to researchers working on gamma-distributed variables.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives parametrizations that turn the sum of two linearly correlated chi-squares into a gamma and the difference into a variance-gamma, with extensions to multiple gammas, but leaves the accuracy range for correlation and degrees of freedom unstated.

read the letter

The central point is that this short note derives reductions of the sum and difference of linearly correlated chi-squared variables to gamma and variance-gamma distributions respectively, using convenient parametrizations. It also shows how the sum case extends to several chi-squares with different parameters and carries over directly to gamma variables that share scale or shape and scale. That mapping is the actual new piece here, and it looks like a straightforward extension of known families rather than a deep re-derivation from scratch. The simulation checks are presented as fitting well, which supplies the main empirical support for the approximations being usable in practice. For someone who already works with gamma models and occasionally needs quick distributional handles on sums or differences, these forms could cut down on custom moment calculations or numerical work. The main limitation is the missing bounds. The abstract and the stress-test note both flag that no explicit conditions appear on the correlation strength or the degrees of freedom, yet the moment-matching step to a single gamma is known to degrade when correlation is high or nu is small. Without reported error curves or validity ranges, a reader has to test the approximations themselves outside moderate regimes. The derivations themselves do not look circular from the description. This is a methods note aimed at applied statisticians who model with gamma variables and want ready-made approximations rather than a broad theoretical audience. It is concrete enough and has enough simulation backing to warrant sending to peer review in a statistical methods journal, even if the referee will likely ask for clearer statements on the operating range.

Referee Report

2 major / 1 minor

Summary. The manuscript derives approximate distributions for the sum and difference of two linearly correlated chi-squared random variables (same degrees of freedom). It claims these reduce to a gamma distribution (sum) and a Variance-Gamma distribution (difference), with extensions to sums of multiple gamma variables sharing a common scale and to differences of gamma variables sharing common shape and scale. The approximations are tested via simulation and described as showing remarkable fits; the results are positioned as useful for researchers working with gamma-distributed variables.

Significance. If the claimed reductions hold with controlled error across a documented range of correlation strengths and degrees of freedom, the approximations would supply convenient, closed-form parametric forms for sums and differences of correlated gammas. The noted generalization to multiple chi-squared variables is a constructive feature that could broaden applicability in statistical modeling.

major comments (2)

[Abstract / derivation] Abstract and derivation sections: the central claim that the sum 'can be reduced to' a gamma distribution (and the difference to Variance-Gamma) is stated without explicit validity conditions on the correlation coefficient ρ or degrees of freedom ν. Because the characteristic function of the sum is a product over the eigenvalues of the 2×2 correlation matrix, moment-matching accuracy is known to degrade for small ν or |ρ| near 1; the absence of these restrictions makes the scope of the approximation load-bearing for the stated result.
[Simulation section] Simulation validation: the abstract reports 'remarkable results' on simulated data but supplies no quantitative error metrics (e.g., Kolmogorov-Smirnov distances, relative moment errors), no table of tested (ρ, ν) pairs, and no range of validity. Without these, it is impossible to assess whether the gamma/Variance-Gamma fits support the central claim outside narrow regimes.

minor comments (1)

Notation for the correlation structure (linear correlation vs. ρ²) and the precise parametrization of the target gamma and Variance-Gamma distributions should be stated explicitly once in the main text rather than left implicit from the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the scope and validation of our approximations. We address each point below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

Referee: [Abstract / derivation] Abstract and derivation sections: the central claim that the sum 'can be reduced to' a gamma distribution (and the difference to Variance-Gamma) is stated without explicit validity conditions on the correlation coefficient ρ or degrees of freedom ν. Because the characteristic function of the sum is a product over the eigenvalues of the 2×2 correlation matrix, moment-matching accuracy is known to degrade for small ν or |ρ| near 1; the absence of these restrictions makes the scope of the approximation load-bearing for the stated result.

Authors: We agree that the manuscript would benefit from explicit discussion of validity conditions. The derivations rely on moment matching, so accuracy is inherently parameter-dependent. In the revision we will add a dedicated subsection stating the ranges of ρ and ν for which the approximations are reliable (supported by the existing simulations), and we will note the expected degradation as |ρ| approaches 1 or ν becomes small. revision: yes
Referee: [Simulation section] Simulation validation: the abstract reports 'remarkable results' on simulated data but supplies no quantitative error metrics (e.g., Kolmogorov-Smirnov distances, relative moment errors), no table of tested (ρ, ν) pairs, and no range of validity. Without these, it is impossible to assess whether the gamma/Variance-Gamma fits support the central claim outside narrow regimes.

Authors: We concur that quantitative error metrics and a tabulated summary of tested parameters would make the validation more rigorous. The revised manuscript will include a table of (ρ, ν) pairs examined, Kolmogorov-Smirnov distances, relative moment errors, and an explicit delineation of the validity range derived from those results. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations independent of inputs

full rationale

The paper presents derivations reducing sums and differences of correlated chi-squared variables to gamma and Variance-Gamma parametrizations. No self-citations, self-definitional steps, fitted inputs renamed as predictions, or ansatzes smuggled via prior work appear in the abstract or description. The central claims rest on characteristic function products and moment-matching that are presented as independent approximations, with no reduction to the paper's own inputs by construction. This is the expected non-finding for a self-contained derivation note.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Abstract-only review prevents full enumeration; the claimed reductions likely rest on moment-matching or characteristic-function approximations that introduce free parameters for shape and scale, plus standard assumptions on chi-squared properties.

free parameters (2)

gamma shape and scale parameters
Convenient parametrization for the sum approximation is chosen to match the correlated chi-squared behavior; exact fitting procedure not visible in abstract.
Variance-Gamma parameters
Parameters for the difference approximation are selected for flexibility; no explicit values or selection method given.

axioms (2)

domain assumption Linear correlation between chi-squared variables allows closed-form reduction to gamma/Variance-Gamma
Invoked in the derivation statement; no proof or conditions supplied in abstract.
standard math Chi-squared variables share properties extendable to gamma with common scale
Background fact used to claim generalization beyond the two-variable case.

pith-pipeline@v0.9.0 · 5629 in / 1391 out tokens · 37817 ms · 2026-05-25T17:17:30.175233+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION format.url url empty "" url if FUNCTION article output.bibitem format.authors "author" output.check author format.key output output.year.check new.block format.title "title" output.check new.block crossref missing format.jour.vol output format.article.crossref output.nonnull format.pages output if ne...

work page
[2]

barticle [author] Gunst , Richard F. R. F. Webster , John T. J. T. ( 1973 ). Density functions of the bivariate chi-square distribution . J. Statist. Comput. Simulation 2 275--288 . 10.1080/00949657308810052 0378246 barticle

work page doi:10.1080/00949657308810052 1973
[3]

barticle [author] Joarder , Anwar H. A. H. , Omar , M. Hafidz M. H. Gupta , Arjun K. A. K. ( 2013 ). The distribution of a linear combination of two correlated chi-squared variables . Rev. Colombiana Estad\' st. 36 209--219 . 3242198 barticle

work page 2013
[4]

( 2015 )

barticle [author] Klar , Bernhard B. ( 2015 ). A note on gamma difference distributions . Journal of Statistical Computation and Simulation 85 3708--3715 . 3401104 barticle

work page 2015
[5]

Neumann , J J

barticle [author] Kotz , S S. Neumann , J J. ( 1963 ). On the distribution of precipitation amounts for periods of increasing length . Journal of Geophysical Research 68 3635--3640 . barticle

work page 1963
[6]

( 2004 )

barticle [author] Seneta , Eugene E. ( 2004 ). Fitting the variance-gamma model to financial data . J. Appl. Probab. 41A 177--187 . Stochastic methods and their applications . 10.1239/jap/1082552198 2057573 barticle

work page doi:10.1239/jap/1082552198 2004

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION format.url url empty "" url if FUNCTION article output.bibitem format.authors "author" output.check author format.key output output.year.check new.block format.title "title" output.check new.block crossref missing format.jour.vol output format.article.crossref output.nonnull format.pages output if ne...

work page

[2] [2]

barticle [author] Gunst , Richard F. R. F. Webster , John T. J. T. ( 1973 ). Density functions of the bivariate chi-square distribution . J. Statist. Comput. Simulation 2 275--288 . 10.1080/00949657308810052 0378246 barticle

work page doi:10.1080/00949657308810052 1973

[3] [3]

barticle [author] Joarder , Anwar H. A. H. , Omar , M. Hafidz M. H. Gupta , Arjun K. A. K. ( 2013 ). The distribution of a linear combination of two correlated chi-squared variables . Rev. Colombiana Estad\' st. 36 209--219 . 3242198 barticle

work page 2013

[4] [4]

( 2015 )

barticle [author] Klar , Bernhard B. ( 2015 ). A note on gamma difference distributions . Journal of Statistical Computation and Simulation 85 3708--3715 . 3401104 barticle

work page 2015

[5] [5]

Neumann , J J

barticle [author] Kotz , S S. Neumann , J J. ( 1963 ). On the distribution of precipitation amounts for periods of increasing length . Journal of Geophysical Research 68 3635--3640 . barticle

work page 1963

[6] [6]

( 2004 )

barticle [author] Seneta , Eugene E. ( 2004 ). Fitting the variance-gamma model to financial data . J. Appl. Probab. 41A 177--187 . Stochastic methods and their applications . 10.1239/jap/1082552198 2057573 barticle

work page doi:10.1239/jap/1082552198 2004