The mean absolute deviation of the classical discrete distributions: collapse identities, complete asymptotic expansions, and enveloping series

Neven Elezovi\'c

REVIEW 2 major objections 5 minor 21 references

The mean absolute deviation of the classical discrete distributions: collapse identities, complete asymptotic expansions, and enveloping series

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The mean absolute deviation of each classical discrete law collapses to a single point mass; the three non-binomial cases have complete asymptotics with Bernoulli-polynomial coefficients.

desk verdict A carefully built extension of the binomial mean-deviation expansions to the other three classical laws; the results stand if the imported Stirling-type lemma from the companion papers holds up. read the letter →

arxiv 2608.06232 v1 pith:4TSH7Z4G submitted 2026-08-06 math.PR

Neven Elezović This is my paper · ORCID

classification math.PR MSC 60E0562E2041A6060C0511B6833B15

keywords meanabsolutedeviationPoissondistributionnegativebinomialhypergeometricBernoullipolynomialssizebiasingStirlingseriesenveloping

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the mean absolute deviation about the mean of any of the four classical discrete laws—binomial, Poisson, negative binomial, and hypergeometric—is exactly twice a single point mass of the distribution, located just above the mean. It then shows for the three non-binomial laws that this point mass has a complete asymptotic expansion in inverse powers of the natural parameter, with coefficients given in closed form by Bernoulli polynomials evaluated at the fractional part of the mean. The key point is that the coefficients carry the lattice displacement of the mean exactly, so the expansions remain valid at every parameter value, not only along arithmetic subsequences. At integer means the expansions reduce to sign-alternating series in odd powers, and a single kernel argument shows those series envelope the logarithm of the normalized mean absolute deviation: successive partial sums bracket the true value. If true, this gives uniform, arbitrary-order asymptotics for a central measure of spread in the Poisson, negative-binomial, and hypergeometric regimes, with explicit oscillating corrections.

What carries the argument

The argument runs on two black-box asymptotics imported from [9]: the shifted Appell form of Stirling's series, $\log\Gamma(x+t)\sim(x+t-\tfrac12)\log x-x+\tfrac12\log2\pi+\sum_{n\ge1}(-1)^{n+1}B_{n+1}(t)/(n(n+1))x^{-n}$, and Lemma 2.1, a complete expansion for a quotient of gamma functions with unequal scalings. On top of these, the paper uses a common telescoping identity $(k-\mu)P(k)=g(k)P(k)-g(k+1)P(k+1)$ whose $g$ has the degree of the Katz–Ord ratio, and the size-bias absorption identities that turn the collapse prefactor $2g(\nu)$ into $2\sigma^2$ times a single mass of a shifted law. For the enveloping theorems the load-bearing object is the kernel $\varphi(t)=\frac1t(\frac1{e^t-1}-\frac1t+\frac12)$, whose positive partial-fraction expansion (via the zeta identity $\sum_{k\ge1}2(4\pi^2k^2)^{-n}=(-1)^{n-1}B_{2n}/(2n)!$) gives Proposition 4.3: any exponentially decaying strictly negative $\Delta$ produces a series with partial sums bracketing $\int\varphi\,\Delta$. The Bernoulli polynomials $B_{n+1}(\cdot)$ at the lattice displacement of the mean carry the oscillation in all complete expansions.

What would settle it

Take an integer-mean case, say $X\sim\mathrm{Poi}(10)$, and compute $\log E|X-\lambda|/\sqrt{2\lambda/\pi}$ numerically or through the exact identity $J(\lambda)$; then check whether the partial sums of $\sum_{j\ge1}a_{2j-1}(0)\lambda^{1-2j}$ with $a_{2j-1}(0)=-B_{2j}/((2j-1)2j)$ alternate around the value with the stated strict signs, since a single violation would falsify the enveloping theorem. For a non-integer check, compare $\lambda=10.3$ against the first two multiplicative coefficients $d_1,d_2$ of Theorem 4.1.

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Extended reading notes

Core claim

For $X$ among $\mathrm{Bin}(N,p)$, $\mathrm{Poi}(\lambda)$, $\mathrm{NB}(r,p)$, and $\mathrm{Hyp}(N,K,n)$, with mean $\mu$ and $\nu=\lceil\mu\rceil$, the paper establishes a common telescoping identity $(k-\mu)P(k)=g(k)P(k)-g(k+1)P(k+1)$ with $g(k)$ given by (8), from which $E|X-\mu|=2g(\nu)P(\nu)$. Through size-biasing this takes the uniform form $E|X-\mu|=2\sigma^2P^{\downarrow}\{\nu-1\}$, where $P^{\downarrow}$ is the mass function of the corresponding size-biased law. For the Poisson, negative binomial, and hypergeometric laws, the paper derives complete asymptotic expansions: for the Poisson, $E|X-\lambda|\sim\sqrt{2\lambda/\pi}\,\exp(-\sum_{n\ge1}B_{n+1}(\{\lambda\})/(n(n+1))\,\lambda^{-n})$, with analogous Bernoulli-polynomial expansions in the other two cases. The elementary, non-Bernoulli tail of the underlying Stirling expansion is removed exactly by the collapse prefactor $g(\nu)$, so the complete series is pure Bernoulli-polynomial at every order. When the mean is an integer, even-index coefficients vanish and the series becomes a sign-alternating odd series; a single kernel identity shows the series is enveloping, so each partial sum is a strict two-sided bound on the normalization.

Load-bearing premise

The central assumption is that an imported asymptotic formula for ratios of gamma functions with unequal scalings (Lemma 2.1) is valid uniformly over the compact ranges of shifts and scalings used here; if that formula fails or lacks the stated uniformity, the complete expansions (15), (21), and (28) do not follow.

Editorial extensions

If this is right

At integer means each law has a sign-alternating enveloping series, so partial sums provide strict two-sided bounds on the mean absolute deviation.
The elementary tail of the Stirling expansion is cancelled exactly by the collapse prefactor, so no separate non-Bernoulli terms appear at any order in the three non-binomial expansions.
The negative-binomial expansion's effective parameter is $1/(rq)$, equivalently $1/(p\mu)$, so the approximation degrades as $q\to0$; the matching to the Poisson case as $rq\to\lambda$ is left open.
The hypergeometric expansion is naturally written in terms of $Nad$, the independence-table cell mass, rather than the variance $\sigma^2$, and the finite-population factor $N/(N-1)$ never enters the coefficients.
The universal form $E|X-\mu|=2\sigma^2P^{\downarrow}\{\nu-1\}$ gives an exact lattice version of the continuous formula ‘mean deviation equals twice the variance times a density value at the mean’ for all four laws.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One could press the same telescoping scheme on other members of the Pearson/Ord family whose ratio has degree two; the logarithmic distribution, which the paper notes lies outside its framework, is a natural first candidate.
The three open matchings (binomial-to-Poisson as $p=\lambda/N$, negative-binomial-to-Poisson as $rq\to\lambda$, and hypergeometric-to-binomial as $\eta\to0$) could plausibly be resolved by uniform expansions built from the same gamma-quotient lemma, yielding one transition-valid formula instead of the current degradation of the effective parameter.
Because the enveloping series carry strict sign information, they can be used as certified interval arithmetic for the mean absolute deviation: truncating at an odd or even order gives a deterministic upper or lower bound over a whole parameter range.
The continuous Cesàro vanishing of each Poisson coefficient suggests that for generic $\lambda$ the oscillating corrections randomize over a period, so the leading Gaussian term is the honest average; this is a testable high-precision prediction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper completes a natural programme, and the new asymptotic expansions look right. The only real question is the black-box Lemma 2.1 imported from the author's companion preprints; nothing in the paper proves it. If that lemma is solid, the results stand.

What is new: the complete asymptotic expansions for Poisson (15), negative binomial (21), and hypergeometric (28), with closed Bernoulli-polynomial coefficients carrying the lattice displacement exactly. The enveloping theorems at integer means are also new beyond the binomial case, and the single Binet-kernel argument (Proposition 4.3) is clean and sign-consistent. The collapse identities are classical, and the author says so; the telescoping is known in the Pearson/Ord w-function literature. The genuinely useful addition there is the uniform statement (Theorem 3.3) and the size-bias interpretation, which explains why the hypergeometric needs two tail cancellations. The proof assembly is detailed, and I checked the algebraic steps: the cancellation between the elementary tails and the collapse prefactors works, and the coefficients in (21), (22), (28), (29) are consistent.

The soft spot is real but narrow. Section 2 states Lemma 2.1 and the shifted Stirling form (4) as black boxes from [9]. Every complete expansion in the paper goes through Lemma 2.1, so the central theorems are conditional on that lemma's uniformity in the shifts. The author states the uniformity condition explicitly, and the shifts h remain in [0,1), which is a compact set, so the stated condition is exactly what is needed. But there is no proof in this paper, and [9] is a preprint. That is a missing support, not an error I can point to. The rest of the proof chain -- the telescoping, the size-bias cancellations, the Binet-kernel enveloping -- is internally consistent, and I found no sign of circularity or fitted parameters.

The paper also says clearly that matching across the degenerate limits (rq -> lambda, etc.) is left open; that is a scope limitation, not a flaw.

Bottom line: this deserves a serious referee. I would ask the author to either include a proof of Lemma 2.1 or make the companion preprint impossible to miss in the review package; the referee should verify the uniformity claim. The paper is not a desk reject. I would cite it if I worked on asymptotics of discrete laws.

Referee Report

2 major / 5 minor

Summary. The paper develops a unified treatment of the mean absolute deviation about the mean for the binomial, Poisson, negative binomial and hypergeometric laws. Section 3 proves a common telescoping identity, derives the collapse formula E|X−µ| = 2g(ν)P(ν), and recasts it in the universal size-biased form (13). Sections 4–6 derive complete asymptotic expansions for the three non-binomial laws, with Bernoulli-polynomial coefficients carrying the lattice displacement of the mean, and prove enveloping sign-alternating series at integer means through a single Binet-kernel argument. Section 8 gives careful historical attribution. The main results are Theorems 4.1, 5.1, 6.1 and the enveloping Theorems 4.4, 5.3, 6.3.

Significance. If the asymptotic results hold, they are a genuine advance: all-order expansions with closed Bernoulli-polynomial coefficients, exact lattice oscillation, no fitted parameters, and a common size-bias mechanism for the cancellation of elementary tails. The telescoping proof of the classical collapse formulas is clean, the universal form (13) is illuminating, and Proposition 4.3 is an elegant self-contained kernel argument that yields bracketing for all three laws at once. The paper is honest about the classical provenance of the closed forms. The main weakness is that the complete expansions all pass through Lemma 2.1 and Eq. (4), imported as black boxes from the companion preprints [9,10]; the central asymptotic claims are therefore conditional on an unverifiable supporting lemma.

major comments (2)

[§2 (Lemma 2.1 and Eq. (4))] The complete expansions of Theorems 4.1, 5.1 and 6.1 are all obtained by applying the imported gamma-quotient expansion Lemma 2.1 and the shifted Stirling series (4), and neither is proved in this manuscript. This is load-bearing because the required uniformity in the lattice displacement h (which varies with r or N in [0,1)) and in the scalings governed by p or (κ,η) is precisely what legitimizes the Bernoulli-polynomial coefficients to all orders. Please supply a proof of Lemma 2.1 and of the uniform remainder in Eq. (4), or replace the black-box reference by a complete and verifiable derivation; as it stands, the main asymptotic theorems remain conditional.
[§5.1, Step 4 (and §6.1, Step 4)] In the tail-cancellation step, the paper truncates the exponent of P(ν) after M series terms, replaces the truncated elementary tail by the exact log-factor via Lemma 2.2, and then multiplies by the growing prefactor 2ν/p (respectively 2g(ν)). The argument asserts that after cancellation the relative error remains O(r^{-M-1}) (respectively O(N^{-M-1})) uniformly in h, but the interaction of a prefactor of order r with the uniform remainder and with the boundedness of the factor pν/(qr) is not shown in detail. This uniformity is exactly what Lemma 2.1 must deliver, so the check should be made explicit.

minor comments (5)

[§5.1, proof of Theorem 5.1, Step 2] The displayed equality 'p√2πrq' should read 'p/√(2πrq)' or 'p(2πrq)^{-1/2}'; as printed the equality to 1/√(2πσ²) is false.
[§2 (Lemma 2.1)] The lemma is stated for shifts in compact sets, but the applications require h ∈ [0,1); to apply the lemma verbatim one should note that [0,1] is compact and that the uniformity extends to the endpoint by the stated remainder estimates.
[§7 (family table)] For the binomial, write the weight with explicit parentheses as p^{-m}+(-1)^{m+1}q^{-m} to avoid a possible parse error; the text already notes that the negative-binomial entry (p/q)^m-p^m is the same bracket in reverse order, which is helpful and should remain.
[Theorem 4.1] The uniformity statement 'uniform in θ∈[0,1)' should explicitly tie θ to λ through θ={λ}; otherwise the statement can be misread as uniformity over an arbitrary independent parameter θ, which is not what the proof establishes.
[References] Since [9] and [10] are companion preprints that are not yet published, the paper should state their availability or review status; this would help readers verify the imported Lemma 2.1 and Eq. (4).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the target mean-deviation expansions are derived from general, target-free asymptotic lemmas, with no fitted parameters or definitional equivalences.

full rationale

The derivation chain shows no circular reduction. The collapse identities of Theorem 3.1 are verified by direct algebraic substitution from the explicit mass functions, and the resulting closed forms are explicitly credited to classical sources (Ramasubban, Crow, Katti, Kamat; Diaconis–Zabell). The three complete expansions (15), (21), and (28) are assembled from two imported results, Eq. (4) and Lemma 2.1, which the paper honestly labels as black boxes in Section 2. Those results are general analytic statements about shifted Stirling series and balanced gamma quotients; their stated assumptions do not include mean absolute deviation, and neither result contains the target expansions. The target coefficients emerge only after the paper's own cancellation steps (size-bias prefactors and Lemma 2.2). No parameter is fitted and then renamed a prediction; the Bernoulli-polynomial coefficients are closed-form functions of the lattice displacement h, which is itself an exact function of the law's parameters. The enveloping theorems are proved independently via Binet's kernel and Proposition 4.3, with the moment coefficients c_n computed directly from the relevant exponential combinations. The only caveat is that Eq. (4) and Lemma 2.1 are imported from the author's companion preprint [9] without proof here; that is a verification/correctness risk, not circularity, because the imported lemmas are target-free and parameter-free in their assumptions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters and no invented entities are present. The paper derives its results from standard asymptotic identities; two supporting lemmas are taken from the author's companion preprints rather than proved here.

assumptions (5)

standard math Shifted Appell form of Stirling's series, Eq. (4), with remainder uniform in the shift t on compact sets.
Imported from companion preprint [9, Section 2]; used in Theorems 4.1, 5.1, 6.1 and the enveloping identities.
standard math Gamma-quotient expansion Lemma 2.1, Eq. (5), with scalar balance Lambda = 0 and Bernoulli-polynomial coefficients S_{n+1}.
Imported from [9, Lemma 2.1]; the backbone of all three complete expansions.
standard math Binet's first formula (17) and the Mittag-Leffler expansion of the kernel phi(t).
Used in Proposition 4.3 and in the enveloping theorems to obtain exact integral representations.
standard math Bernoulli polynomial identities (2): the difference identity and the reflection identity.
Used to split elementary tails from the Appell part and to evaluate coefficients at integer means.
domain assumption Non-degeneracy and parameter-compactness assumptions: 0<p<1, lambda>0, r>0, 1<=K<=N-1, 1<=n<=N-1; for asymptotics, p in a compact subset of (0,1) and margins kappa, eta in a compact subset of (0,1)^2.
Keeps the size-bias-shifted laws and the scaling parameters in compact sets, which the imported gamma-quotient lemma requires.

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Cite this review

Pith. "Pith review of The mean absolute deviation of the classical discrete distributions: collapse identities, complete asymptotic expansions, and enveloping series." pith.science (2026). https://pith.science/paper/4TSH7Z4G

@misc{pith2026260806232,
  author       = {Pith},
  title        = {Pith review of: The mean absolute deviation of the classical discrete distributions: collapse identities, complete asymptotic expansions, and enveloping series},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4TSH7Z4G}},
  note         = {Machine review of arXiv:2608.06232}
}

abstract

For each of the four classical discrete laws --- binomial, Poisson, negative binomial and hypergeometric --- the mean absolute deviation about the mean collapses to a single point mass. We give a common telescoping proof of these identities and interpret the resulting closed forms by size biasing. We then derive complete asymptotic expansions for the Poisson ($\lambda\to\infty$), negative binomial ($r\to\infty$, $p$ fixed) and hypergeometric ($N\to\infty$, margins in fixed proportion) cases, extending the binomial expansion from the companion papers. The coefficients are given in closed Bernoulli-polynomial form and carry the lattice displacement of the mean exactly. At integer means the expansions reduce to sign-alternating odd series, and a single Binet-kernel argument shows that these series envelop the logarithm of the normalised mean absolute deviation: successive partial sums bracket it.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [9]

Local binomial expansions with an Appell shift, and the mean absolute deviation of the binomial distribution

N. Elezović,Local binomial expansions with an Appell shift, and the mean absolute deviation of the binomial distribution, preprint,arXiv:2607.18494 [math.CA], 2026

work page Pith review arXiv 2026

[2] [1]

Afendras, N

G. Afendras, N. Papadatos, V. Papathanasiou,An extended Stein-type co- variance identity for the Pearson family with applications to lower variance bounds, Bernoulli17(2011), no. 2, 507–529

work page 2011

[3] [2]

J. Ai, C. Pelekis,On lower bounds for hypergeometric tails, preprint, arXiv:2601.09485, 2026

work page Pith review arXiv 2026

[4] [3]

Arratia, L

R. Arratia, L. Goldstein, F. Kochman,Size bias for one and all, Probab. Surv.16(2019), 1–61

work page 2019

[5] [4]

G. E. Bardwell,On certain characteristics of some discrete distributions, Biometrika47(1960), 473–475

work page 1960

[6] [5]

R. P. Brent,Asymptotic approximation of central binomial coefficients with rigorous error bounds, Open J. Math. Sci.4(2020), no. 1; arXiv:1608.04834

work page Pith review arXiv 2020

[7] [6]

Cacoullos, V

T. Cacoullos, V. Papathanasiou,Characterizations of distributions by vari- ance bounds, Statist. Probab. Lett.7(1989), no. 5, 351–356

work page 1989

[8] [7]

E. L. Crow,The mean deviation of the Poisson distribution, Biometrika45 (1958), 556–562

work page 1958