bibkey: meerweeks2024boundedmoments
authors: David J. Meer; Eric R. Weeks
year: 2024
title: "Bounded distributions place limits on skewness and larger moments"
doi: 10.1371/journal.pone.0297862
url: https://arxiv.org/abs/2308.05006v2
claim: "The paper proves lower bounds on the skewness of distributions bounded on one side in terms of the coefficient of variation, derives the corresponding bidisperse limits for all standardized moments D_n, conjectures that the bidisperse limit D_5 ≥ z^3 - z + z^(-1) - z^(-3) holds for every distribution bounded from below (and its analogue for distributions bounded from above), and suggests that the two-sided bidisperse kurtosis limits hold for every distribution bounded on both sides."
strata_touched:
  - D5/S3/Analytic/MomentMinors/BoundedFifthMomentLimit
  - D5/S3/Analytic/MomentMinors/BoundedKurtosisLimitRefutation
license: citation-only
triage: anchor

Bounded distributions place limits on skewness and larger moments

D. J. Meer and E. R. Weeks, arXiv:2308.05006v2 [stat.AP, cond-mat.stat-mech]; PLOS ONE 19, e0297862 (2024). Quotations are from the v2 TeX source.

The normalization for distributions bounded from below:

Now consider the general case of a distribution $P(x)$ bounded by $x_{\min }$ from below and with a mean $\mu$ which might be zero. Let us assume $P(x)$ has a nontrivial domain, which is to say, it is not a distribution which is only nonzero at one value (which would thus be $\sigma =0,D_3=0$). $\delta =\sigma /(\mu -x_{\min }).$

(the display is written inline here).

The conjecture:

One can immediately put in a value for $a$ of interest and get a potential limit of $D_n$. For example, for distributions bounded from below by $x_{\min }$ we conjecture $D_5\ge D_{5,\min }=z^3-z^1+z^{-1}-z^{-3}$ with $z=\delta /(1-x_{\min }/\mu )$ as above. As with $D_3$, our conjectured $D_{5,\max }$ is a similar equation using $z=\delta /(x_{\max }/\mu -1)$.

(the display is written inline here).

The two-sided kurtosis limits, after the definitions of ${\delta }_1$, ${\delta }_2$, $D_{4,m}(\delta )={\left(\frac{\delta }{{\delta }_m}-\frac{{\delta }_m}{\delta }\right)}^2+1$ and ${\delta }^{\prime }=\sqrt{{\delta }_1{\delta }_2}$:

The limits on kurtosis $D_4$ are then $1\le D_4\le D_{4,2}(\delta )$ $(\delta <{\delta }_1)$, $D_{4,1}(\delta )\le D_4\le D_{4,2}(\delta )$ $({\delta }_1\le \delta <{\delta }^{\prime })$

(the display is written inline here), and:

While we have not proven that the bidisperse distribution sets the limits for $D_4$ for all other distributions, this is suggestive that Inequalities \ref{D4mostgeneral} are indeed limits for the kurtosis for any distribution.

Verified locator

Source: https://arxiv.org/abs/2308.05006v2

• URL: https://arxiv.org/abs/2308.05006v2 (source plos_latex_template.tex, md5 4452c28a26c7a76c87642af46fe0c8a9):
  • the moments $m_n$ and $D_n$, l. 228–236;
  • the coefficient of variation $\delta =\sigma /\mu$, l. 290–293;
  • the bounded-below and bounded-above normalizations, l. 396–408;
  • ${\delta }_0$, the two-sided quantities ${\delta }_1,{\delta }_2,D_{4,m},{\delta }^{\prime }$ and the kurtosis limits, Eq. D4mostgeneral, l. 437–467;
  • Eq. DNlimits and the conjecture, l. 496–510;
  • the remark on counterexamples for $n=5$, l. 532.
• DOI: 10.1371/journal.pone.0297862 (https://doi.org/10.1371/journal.pone.0297862; Crossref lists PLOS ONE, volume 19, article e0297862, published 9 February 2024).