REVIEW 3 major objections 6 minor 18 references
The classical discrete laws near the mode: complete local expansions for the negative binomial and hypergeometric distributions
T0 review · 3 major / 6 minor · reviewed 2026-08-27 · deepseek-v4-flash
Pith's one-line read The paper proves complete local expansions near the mode for the negative binomial and hypergeometric laws, with closed Bernoulli coefficients, and shows the first-order term yields the exact mode.
desk verdict Complete Bernoulli-form local expansions for the negative binomial and hypergeometric laws, with a sharp mode-vertex discrepancy bound; the main theorems hinge on a standard gamma-quotient lemma quoted from a companion paper rather than proved here. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the balanced gamma quotient expansion: if $\sum_j \lambda_j = \sum_k \mu_k$, then the logarithm of $\prod_j \Gamma(\lambda_j x+u_j)/\prod_k \Gamma(\mu_k x+v_k)$ expands as $\Theta x + U\log x + C$ plus a series whose $n$-th coefficient is a combination of Bernoulli polynomials of the shifts $u_j,v_k$ with reciprocal powers of the scalings $\lambda_j,\mu_k$ (Eq. (4) of the paper). This identity, together with the shifted Stirling series for $\log\Gamma$, converts the gamma-factor representation of each probability mass function into a complete Bernoulli-coefficient expansion. The negative binomial uses three gamma factors with scalings $1/p,1,q/p$; the hypergeometric uses nine factors arranged along the $2\times 2$ table with margins $K,N-K,n,N-n$, whose independence identity $ad=bc$ assembles the constants. The reciprocal Bernoulli polynomials $\widehat B_n(t)$, generated by $\tfrac{z}{2}\coth\tfrac{z}{2}e^{zt}$, carry the even normalisation, and the quadratic part of the first coefficient supplies the mode vertex.
What would settle it
Take a fixed parameter compact and a fixed truncation order $M$, compute the exact log mass for many large $r$ (or $N$) and displacements $t$, and subtract the truncated expansion; the remainder must be bounded by a constant times $r^{-M-1}$ (or $N^{-M-1}$) uniformly. A single sequence of parameters within the stated range where the remainder fails to decay at that rate would refute the main theorems; likewise, checking exact equality of the discrepancy identity for a few parameter values is a direct test of the mode theorem.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is that for $X \sim \mathrm{NB}(r,p)$ and $X \sim \mathrm{Hyp}(N,K,n)$, the logarithm of the normalised point mass admits a complete expansion in integer powers of $1/r$ or $1/N$, uniformly for fixed bounded displacement $t$ from the mean. For the negative binomial, $\log\!\bigl(\sqrt{2\pi\sigma^2}\,P(\mu+t)\bigr) \sim \sum_{n\ge 1} A_n(t;p)/r^n$ with $A_n = \frac{(-1)^{n+1}}{n(n+1)}\bigl[p^n B_{n+1}(t) - (p/q)^n B_{n+1}(t+1) - B_{n+1}\bigr]$; for the hypergeometric, an analogous nine-gamma assembly along the $2\times 2$ table gives coefficients in terms of reciprocal powers of the cell proportions. The paper's own emphasis is that the coefficients are closed-form in terms of Bernoulli polynomials, that they carry the exact lattice displacement, and that the even normalisation replaces each $B_{n+1}$ with the reciprocal Bernoulli polynomial $\widehat B_{n+1}$. The first-order coefficient then yields the mode: in the negative binomial case the identity $\mu+t_c+\tfrac12=(r-1)q/p$ is exact, and in the hypergeometric case the discrepancy $\mu+t_c+\tfrac12-(n+1)(K+1)/(N+2)=-(2K-N)(2n-N)/(N^2(N+2))$ is proved to be smaller than $1/(N+2)$, hence below the lattice resolution of the threshold, so the nearest-integer-to-vertex rule selects the exact mode, with ties exactly at the two-mode configurations.
Load-bearing premise
The entire coefficient machinery rests on the balanced gamma quotient expansion with unequal scalings, stated as Eq. (4) and taken from a companion lemma rather than proved here; if that expansion or its uniform remainder were not valid, Theorems 4.1 and 5.1 would not be established.
Editorial extensions
If this is right
- For any fixed truncation order $M$, the point mass at $k=\mu+t$ differs from the first $M$ terms by $O(r^{-M-1})$ uniformly over parameter compacts, so the approximation can be used at any finite precision.
- For the negative binomial, the vertex identity $\mu+t_c+\tfrac12=(r-1)q/p$ is exact, so the largest mode is $\lfloor (r-1)q/p\rfloor$, with two modes exactly when that number is an integer.
- For the hypergeometric law, the nearest-integer-to-vertex rule gives the classical mode $\lfloor (n+1)(K+1)/(N+2)\rfloor$ whenever the threshold is not an integer; when it is an integer, the rule selects one of the two tied modes.
- The even normalisation replaces every Bernoulli polynomial of the displacement by a reciprocal Bernoulli polynomial, so the reflection symmetries of the laws appear as exact symmetries of the coefficients.
- Along the seam $r\to\infty$, $q\to 0$, $rq=\lambda$ fixed, the negative binomial expansion converges coefficientwise to the Poisson expansion.
Reading between the lines
- A testable extension is to apply the same balanced gamma quotient to a general contingency table with fixed margins; the paper's hypergeometric expansion is the $2\times2$ instance, and a multivariate generalisation would follow the same cell-proportion grammar.
- The theorem suggests a general criterion: an asymptotic first-order vertex that misses a rational lattice threshold by less than the reciprocal of the threshold's denominator still yields the exact integer rule; checking this on other lattice laws would show how widely the phenomenon holds.
- Because the discrepancy $-(2K-N)(2n-N)/(N^2(N+2))$ is, up to sign, the numerator of the classical hypergeometric skewness, one could test whether a skewness-corrected vertex recovers the threshold exactly rather than merely within one lattice step.
- The closed Bernoulli coefficients could support explicit error bounds and continuity corrections for the hypergeometric law, extending refinements already available for the negative binomial.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops complete local asymptotic expansions for the probability mass function of the negative binomial distribution as r→∞ and of the hypergeometric distribution as N→∞, with the argument written as k=μ+t for bounded lattice displacements t. Theorems 4.1 and 5.1 give all-order expansions of log(√(2πσ²)P(μ+t)) in powers of 1/r and 1/N, with coefficients in closed Bernoulli-polynomial form that retain the exact lattice shift. Theorems 4.2 and 5.2 give the corresponding 'even' normalisations, in which reciprocal Bernoulli polynomials replace Bernoulli polynomials, and identify reflection symmetries. Theorems 4.4 and 5.3 use the first-order coefficient to derive mode rules: for the negative binomial the vertex is exact, while for the hypergeometric the vertex misses the classical threshold by an explicit small amount that is always below the lattice resolution, so the nearest-integer rule still selects the exact mode. A coefficientwise Poisson-degeneracy check is given in Proposition 4.6.
Significance. If the results are fully established, they provide the first complete asymptotic expansions in the literature for these two classical lattice laws in the mode regime, with explicit closed coefficients rather than cumulant-based Edgeworth expansions. The mode-vertex identities, especially the exact hypergeometric discrepancy (18) and the sub-resolution bound, are elegant and genuinely new. The proofs are mostly explicit, the mode statements are independently verifiable by direct ratio comparisons, and the Poisson degeneracy check provides a useful internal consistency test. The main weakness is that the central expansion engine, Eq. (4), is imported from an unpublished companion preprint and is not proved here; this is a verification and self-containedness problem rather than an evident internal inconsistency.
major comments (3)
- [Section 2, Eq. (4)] The proofs of Theorems 4.1 and 5.1, and consequently of Theorems 4.2 and 5.2, are built on the balanced gamma-quotient expansion (4), which is attributed to Lemma 2.1 of the companion preprint [2] and is not proved in this manuscript; moreover, the constants Θ, U, C in (4) are not stated here, only referenced. Since every Bernoulli-polynomial coefficient in (7) and (14) comes from the series part of (4), and the prefactors are assembled from its elementary constants, this is a load-bearing external dependency. The mode theorems are less vulnerable because they are supplemented by direct ratio computations, but the paper's central claim of complete expansions is not self-contained. Please provide a proof of (4) or replace the citation with a published source that includes the same uniform remainder, and state the constants Θ, U, C explicitly.
- [Sections 4.1 and 5.1 (proofs of Theorems 4.1 and 5.1)] The assembly of the normalisation prefactor is delegated to [4, §5] and [4, §6], respectively. In particular, the cancellation of the Θx term, the claim U = -1/2, and the identities e^C q^t r^{-1/2} = (2πσ²)^{-1/2} and e^C N^{-1/2} = (2πNad)^{-1/2} are asserted without derivation. Because Corollary 4.5 evaluates the expansion at the mode and depends on the numerical prefactor, these normalisations are not merely cosmetic. Please either include the elementary-constant computation in the proofs or collect it in a self-contained appendix.
- [Section 2, remainder statements] The paper states that 'exponentials of truncated series are re-expanded through the standard Bell recursion' and that the mass itself has a complete expansion of the same shape, but no formal lemma quantifies the resulting remainder for the mass after exponentiation. This is standard and likely fixable, but as written it is an unproved step in the claim that the expansions are complete for the probabilities P(k), not only for their logarithms.
minor comments (6)
- [Section 4.1, proof of Theorem 4.1] The displayed formula 'eCqtr−1/2' is garbled; it should read e^C q^t r^{-1/2}.
- [Section 5.1, proof of Theorem 5.1] The displayed formula 'e CelN−1/2' is garbled; it should read e^C N^{-1/2}.
- [Corollary 4.5] The fractional-part notation f={(r−1)q/p} is used without definition; please define {x} as the fractional part of x.
- [Equation (11)] There is a stray colon after O(r^{-2}) in the displayed formula; this appears to be a typographical artifact.
- [Section 6, table] The table entry for the Poisson law is potentially misleading: the Poisson expansion (6) involves B_{n+1}(t+1), not the generic B_{m+1}(t) form described in the preceding sentence. Please clarify the convention used in the table.
- [Sections 1 and 4.2] The term 'entropy normalisation' is used before the relevant prefactor has been defined; consider introducing the phrase 'local entropy prefactor' at its first use in Theorem 4.2 and Theorem 5.2.
Circularity Check
No circularity: the central expansions follow from a standard gamma-quotient lemma with independent classical provenance, and the mode claims are proved by direct ratio computations.
full rationale
The derivation chain is not circular. Theorems 4.1 and 5.1 expand log normalized probabilities by applying the balanced gamma-quotient expansion (4) to the gamma-factor representation of each pmf; the coefficients (7) and (14) come from the Bernoulli-polynomial series part of that lemma, and the prefactor constants from elementary assembly using the balance condition and the independence identities. Equation (4) is not defined in terms of the negative binomial or hypergeometric laws; it is a general, parameter-free expansion of a log gamma quotient under equal total scalings, with a uniform remainder, and the paper explicitly anchors it to the shifted Stirling series (3), attributing that series to DLMF (5.11.8) and to Tricomi-Erdelyi [18]. Thus the self-citation to [2, Lemma 2.1] is a reference to an independent, externally checkable classical result rather than an unverified premise that presupposes the paper's conclusions. The mode theorems 4.4 and 5.3 are proved separately by direct ratio computations and exact algebra: the negative binomial mode follows from P(k)/P(k-1) >= 1 iff k <= (r-1)q/p, and the hypergeometric discrepancy identity (18) is an exact algebraic consequence of the explicit quadratic form of A1. No fitted parameters, no data subsets, and no renamed empirical predictions appear anywhere. The Poisson-seam check (Proposition 4.6) provides an external consistency test against the known Poisson expansion. The only arguable weakness is that Eq. (4) is stated rather than proved in this paper, but that is a verification or self-containedness concern, not circularity, because the cited lemma is classical and not equivalent to the target results. Hence no circular step can be exhibited, and the score is 0.
Assumptions & free parameters
assumptions (5)
- standard math Shifted Stirling series for log Gamma, Eq. (3), from DLMF 5.11.8.
- standard math Balanced gamma quotient expansion with unequal scalings, Eq. (4), Lemma 2.1 of [2].
- domain assumption Katz-Ord classification that probability ratios of the four laws are rational of degree one or two.
- domain assumption Uniformity hypotheses: p in compact subsets of (0,1), kappa and eta in compact subsets of (0,1)^2, t in a fixed compact.
- standard math Bernoulli polynomial identities and reciprocal Bernoulli generating function, Eqs. (1) and (2).
Cite this review
Pith. "Pith review of The classical discrete laws near the mode: complete local expansions for the negative binomial and hypergeometric distributions." pith.science (2026). https://pith.science/paper/BWJ5CKED
@misc{pith2026260813631,
author = {Pith},
title = {Pith review of: The classical discrete laws near the mode: complete local expansions for the negative binomial and hypergeometric distributions},
year = {2026},
howpublished = {\url{https://pith.science/paper/BWJ5CKED}},
note = {Machine review of arXiv:2608.13631}
}
read the original abstract
We derive complete local asymptotic expansions near the mode for the negative binomial and hypergeometric laws, complementing the binomial expansion obtained in the companion papers. The coefficients are given in closed Bernoulli-polynomial form and retain the exact lattice displacement from the mean. For each law an entropy normalisation replaces the Bernoulli polynomials by reciprocal Bernoulli polynomials and makes the natural reflection symmetry visible. The first-order coefficient also recovers the exact mode rule: in the Katz cases the vertex gives the classical threshold exactly, while in the hypergeometric case its explicit discrepancy is always too small to change the selected integer mode.
Reference graph
Works this paper leans on
-
[2]
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
-
[1]
NIST Digital Library of Mathematical Functions, Release 1.2,https:// dlmf.nist.gov/
-
[3]
N. Elezović,Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion, preprint, arXiv:2607.19844[math.PR], 2026
work page Pith review arXiv 2026
-
[4]
N. Elezović,The mean absolute deviation of the classical discrete distribu- tions: collapse identities, complete asymptotic expansions, and enveloping series, preprint,arXiv:2608.06232[math.PR], 2026
work page Pith review arXiv 2026
-
[5]
N. L. Johnson, S. Kotz, A. W. Kemp,Univariate Discrete Distributions, 2nd ed., Wiley, New York, 1992
1992
-
[6]
Katz,Unified treatment of a broad class of discrete probability distribu- tions, in: G
L. Katz,Unified treatment of a broad class of discrete probability distribu- tions, in: G. P. Patil (ed.),Classical and Contagious Discrete Distributions, Statistical Publishing Society, Calcutta / Pergamon Press, Oxford, 1965, pp. 175–182
work page 1965
-
[7]
B. C. Kellner,Faulhaber polynomials and reciprocal Bernoulli polynomials, Rocky Mountain J. Math.53(2023), 119–151
work page 2023
-
[8]
R. Kaas, J. M. Buhrman,Mean, median and mode in binomial distributions, Statist. Neerlandica34(1980), 13–18
work page 1980
Show all 18 references
-
[9]
Kabluchko, A
Z. Kabluchko, A. Marynych, H. Sulzbach,Mode and Edgeworth expansion for the Ewens distribution and the Stirling numbers, J. Integer Seq.19 (2016), Article 16.8.8
2016
-
[10]
J. E. Kolassa, P. McCullagh,Edgeworth series for lattice distributions, Ann. Statist.18(1990), 981–985
1990
-
[11]
Govindarajulu,Normal approximations to the classical discrete distribu- tions, Sankhy¯ a Ser
Z. Govindarajulu,Normal approximations to the classical discrete distribu- tions, Sankhy¯ a Ser. A27(1965), 143–172
1965
-
[12]
Molenaar,Approximations to the Poisson, Binomial and Hypergeomet- ric Distribution Functions, Mathematical Centre Tract 31, Mathematisch Centrum, Amsterdam, 1970
W. Molenaar,Approximations to the Poisson, Binomial and Hypergeomet- ric Distribution Functions, Mathematical Centre Tract 31, Mathematisch Centrum, Amsterdam, 1970
1970
-
[13]
W. L. Nicholson,On the normal approximation to the hypergeometric dis- tribution, Ann. Math. Statist.27(1956), 471–483. 14
1956
-
[14]
J. K. Ord,On a system of discrete distributions, Biometrika54(1967), 649–656
1967
-
[15]
Ouimet,A refined continuity correction for the negative binomial distri- bution and asymptotics of the median, Metrika86(2023), 827–849
F. Ouimet,A refined continuity correction for the negative binomial distri- bution and asymptotics of the median, Metrika86(2023), 827–849
2023
-
[16]
Ouimet,A precise local limit theorem for the multinomial distribution and some applications, J
F. Ouimet,A precise local limit theorem for the multinomial distribution and some applications, J. Statist. Plann. Inference215(2021), 218–233
2021
-
[17]
Ouimet,On the Le Cam distance between multivariate hypergeometric and multivariate normal experiments, Results Math.77(2022), art
F. Ouimet,On the Le Cam distance between multivariate hypergeometric and multivariate normal experiments, Results Math.77(2022), art. 47
2022
-
[18]
F. G. Tricomi, A. Erdélyi,The asymptotic expansion of a ratio of gamma functions, Pacific J. Math.1(1951), 133–142. 15
1951
Reviewed August 27, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.