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arxiv: 1906.09485 · v1 · pith:W6KYCAW6new · submitted 2019-06-22 · 🧮 math.ST · stat.TH

Relative variation indexes for multivariate continuous distributions on [0,infty)^k and extensions

C\'elestin C.Kokonendji , Aboubacar Y. Tour\'e , Amadou Sawadogo This is my paper

Pith reviewed 2026-05-25 17:43 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords multivariate variation indexesrelative dispersioncontinuous distributionsnon-negative orthantquadratic formsmodel discriminationdispersion indexes
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The pith

Ratios of quadratic forms in the mean and covariance define indexes for departure from uncorrelated exponential distributions.

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

This paper introduces scalar indexes to measure how far any multivariate continuous distribution on the non-negative orthant deviates from a reference distribution such as the uncorrelated exponential model. The indexes are constructed as ratios of two quadratic forms involving only the mean vector and the covariance matrix. They extend relative Fisher dispersion indexes from count models and support discrimination among positive continuous distributions. Generalized, marginal, and relative versions are considered along with their asymptotic properties.

Core claim

We introduce some new indexes to measure the departure of any multivariate continuous distribution on non-negative orthant from a given reference one such the uncorrelated exponential model, similar to the relative Fisher dispersion indexes of multivariate count models. The proposed multivariate variation indexes are scalar quantities, defined as ratios of two quadratic forms of the mean vector and the covariance matrix.

What carries the argument

The multivariate variation index, a scalar ratio of two quadratic forms of the mean vector and covariance matrix, that quantifies departure from the reference distribution.

If this is right

  • These indexes discriminate between continuous positive distributions.
  • Generalized and multiple marginal variation indexes with and without correlation structure are defined.
  • Asymptotic behavior and other properties are studied.
  • Numerical applications lead to appropriate choices of multivariate models.

Where Pith is reading between the lines

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

  • The indexes might apply to model selection tasks involving positive multivariate data in reliability or environmental statistics.
  • If first and second moments prove insufficient, versions incorporating higher moments could be developed as direct extensions.
  • Similar ratio constructions might connect to other moment-based dispersion tools already used in multivariate analysis.

Load-bearing premise

The departure from the reference distribution can be adequately captured by these ratios of quadratic forms involving only first and second moments for the purpose of discriminating between distributions.

What would settle it

Two distributions sharing identical mean vector and covariance matrix but differing in higher moments receive the same index value while exhibiting visibly different departures from the uncorrelated exponential reference.

Figures

Figures reproduced from arXiv: 1906.09485 by Aboubacar Y. Tour\'e, Amadou Sawadogo, C\'elestin C.Kokonendji.

Figure 1
Figure 1. Figure 1: Boxplots for the targets GVI and MVI with 100 replicates according to sample [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Support of bivariate distributions with maximum correlations (positive in blue [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
read the original abstract

We introduce some new indexes to measure the departure of any multivariate continuous distribution on non-negative orthant from a given reference one such the uncorrelated exponential model, similar to the relative Fisher dispersion indexes of multivariate count models. The proposed multivariate variation indexes are scalar quantities, defined as ratios of two quadratic forms of the mean vector and the covariance matrix. They can be used to discriminate between continuous positive distributions. Generalized and multiple marginal variation indexes with and without correlation structure, respectively, and their relative extensions are discussed. The asymptotic behavior and other properties are studied. Illustrative examples and numerical applications are analyzed under several scenarios, leading to appropriate choices of multivariate models. Some concluding remarks and possible extensions are made.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces scalar multivariate variation indexes (generalized, marginal, and relative) defined as ratios of quadratic forms in the mean vector and covariance matrix. These are intended to quantify departure from an uncorrelated exponential reference distribution for continuous multivariate distributions on [0,∞)^k, analogous to relative Fisher dispersion indexes for counts. The paper studies asymptotic behavior and properties, and provides illustrative numerical examples for model discrimination.

Significance. The moment-based construction supplies a simple, explicit scalar measure that can be computed directly from mean and covariance without parameter fitting. The asymptotic analysis and concrete numerical applications under several scenarios constitute a strength, offering a practical tool for comparing distributions that share first and second moments.

major comments (2)
  1. [Abstract and §1] Abstract and introduction: the claim that the indexes 'can be used to discriminate between continuous positive distributions' is not supported in general. Because every index is a ratio of quadratic forms involving only the mean vector and covariance matrix, the value is identical for any two distributions matching on first and second moments, irrespective of higher-moment or tail differences. This is load-bearing for the discrimination purpose stated in the abstract.
  2. [Definition of the indexes (presumably §2)] Definition of the indexes: the construction supplies no mechanism by which higher moments or tail behavior can affect the index values. Consequently the indexes cannot distinguish, e.g., a multivariate gamma from a multivariate lognormal that share the same mean and covariance, undermining the claim that they measure departure from the uncorrelated exponential reference in a manner useful for broad model selection.
minor comments (2)
  1. [Numerical examples] Numerical examples section: state explicitly whether the compared distributions differ only in mean/covariance or also in higher moments, so readers can assess the practical scope of the discrimination.
  2. [Notation and definitions] Notation for the quadratic forms: ensure the matrix expressions are written with consistent dimensions and that the reference exponential parameters are stated explicitly when the relative indexes are introduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We agree that the indexes are defined solely in terms of the mean vector and covariance matrix, and we will revise the abstract and introduction to clarify the scope of their discriminatory power.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and introduction: the claim that the indexes 'can be used to discriminate between continuous positive distributions' is not supported in general. Because every index is a ratio of quadratic forms involving only the mean vector and covariance matrix, the value is identical for any two distributions matching on first and second moments, irrespective of higher-moment or tail differences. This is load-bearing for the discrimination purpose stated in the abstract.

    Authors: We agree with the observation. The indexes are explicitly constructed as functions of the mean vector and covariance matrix only, so they necessarily return identical values for any distributions sharing the same first and second moments. Their utility for discrimination therefore applies to families of models that differ in these moments (as occurs in the numerical examples). We will revise the abstract and §1 to state that the indexes discriminate between continuous positive distributions on the basis of their mean and covariance structures relative to the uncorrelated exponential reference. revision: yes

  2. Referee: [Definition of the indexes (presumably §2)] Definition of the indexes: the construction supplies no mechanism by which higher moments or tail behavior can affect the index values. Consequently the indexes cannot distinguish, e.g., a multivariate gamma from a multivariate lognormal that share the same mean and covariance, undermining the claim that they measure departure from the uncorrelated exponential reference in a manner useful for broad model selection.

    Authors: The referee is correct that the construction contains no dependence on moments beyond the second order. This is intentional and mirrors the relative Fisher dispersion indexes for multivariate counts, which likewise rely only on mean and variance. In the paper the indexes are applied to compare models whose fitted means and covariances differ; we will add an explicit statement in §2 that the indexes quantify departure from the reference in terms of first- and second-moment structure and are therefore most useful when candidate models are distinguished by these moments. revision: yes

Circularity Check

0 steps flagged

No circularity: indexes introduced by explicit definition from moments

full rationale

The paper defines its variation indexes directly as ratios of quadratic forms in the mean vector and covariance matrix (see abstract and full text descriptions of generalized, marginal, and relative indexes). No derivation chain exists in which a claimed result or prediction is shown to equal its inputs by construction, no fitted parameters are relabeled as predictions, and no load-bearing self-citations or uniqueness theorems are invoked to justify the central construction. The proposal is self-contained as a definitional tool for comparing first- and second-moment behavior to a reference distribution.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper's central contribution is the definition of new indexes relying on standard statistical concepts and a chosen reference model; no free parameters are mentioned.

axioms (2)
  • standard math Existence of mean vector and covariance matrix for the distributions
    Indexes defined in terms of these quantities.
  • domain assumption Uncorrelated exponential as appropriate reference model
    Used as the given reference one.
invented entities (1)
  • Multivariate variation indexes no independent evidence
    purpose: To measure departure from reference distribution
    Newly introduced scalar quantities based on quadratic forms.

pith-pipeline@v0.9.0 · 5658 in / 1264 out tokens · 35788 ms · 2026-05-25T17:43:10.957418+00:00 · methodology

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