A binomial-like probability distribution with heavy tails

J\"urgen Gro{\ss}

This paper proposes a mixture of the binomial and a new bi-uniform distribution as a simple heavy-tailed alternative that preserves the binomial mean and fits a classical family sex-ratio data set better than the binomial or beta-binomial.

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2026-08-01 16:08 UTC pith:QQGPMZMB

load-bearing objection A small but honest paper: a new two-parameter mixture that trades flexibility for simplicity, with correct math and a transparent application.

arxiv 2607.18083 v1 pith:QQGPMZMB submitted 2026-07-20 stat.AP

A binomial-like probability distribution with heavy tails

J\"urgen Gro{\ss} This is my paper

classification stat.AP MSC 62E1562F03

keywords binomial distributionbi-uniform distributionmixture distributionheavy tailsoverdispersionsex-ratio dataminimum chi-square estimationgoodness-of-fit

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that a mixture of the binomial distribution and a newly introduced bi-uniform distribution—two flat probability blocks meeting at the integer part of np—is a simple, valid heavy-tailed alternative to the binomial. Because both components share the same mean np, the mixture inherits that mean and its variance is exactly the weighted average of the two component variances. Fitting the single extra weight parameter to a classical data set on sex ratios in families with eight children gives a chi-square statistic of 43.6, clearly below the binomial's 91.9 and the beta-binomial's 53.3. The author presents the construction as an exemplary, low-calculus exercise in building, estimating, and testing a statistical model.

Core claim

The central claim is that the weighted arithmetic mean of the binomial pmf and a bi-uniform pmf produces a binomial-like distribution with heavier tails. The bi-uniform component uses only two probability values, split at the integer part of np, and is constructed to share the binomial's expectation. The mixture therefore has expectation np and variance w·v1 + (1−w)·v2, a simplification that holds only because the components have equal means. The paper proves that the bi-uniform variance v2 always exceeds the binomial variance v1 for n≥2 and 0<p<1, so any positive mixture weight on the bi-uniform guarantees overdispersion. In the classical sex-ratio application, minimum chi-square estimation

What carries the argument

The machinery is the bi-uniform distribution on {0,...,n}: a pmf taking only two values p1 and p2, separated at the breakpoint floor(µ), with µ chosen as np. Its role is to carry excess probability to both tails in a way that is completely specified once µ is fixed. The mixture pX(x)=w·p_binomial(x)+(1−w)·p_biuniform(x) then does the work: the shared mean makes the variance a convex combination, the positive variance difference v2−v1 guarantees heavier tails for any w<1, and w is the single parameter that controls how much tail weight is added.

Load-bearing premise

The load-bearing premise is that the real overdispersion in the data has the same two-block shape as the bi-uniform distribution—constant extra probability on the low side and constant extra probability on the high side, meeting at floor(np); if real excess-tail structure is shaped differently, the fitted weight w will not generalize to other overdispersed data.

What would settle it

Fit the mixture to other overdispersed count data, especially data generated by a beta-binomial with strongly skewed parameters or by a process with tail inflation concentrated at one extreme only. If the optimal weight w moves far from 1 or the chi-square advantage over the beta-binomial disappears, the bi-uniform's two-block tail assumption is not the right generic correction.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Because the mixture preserves the binomial expectation, the method-of-moments estimate of p is unchanged; only the additional weight w is estimated from the variance.
Valid weights require the sample variance to lie between the binomial and bi-uniform variances; outside that range the mixture will not fit, bounding the method's applicability.
Minimum chi-square estimation of w (0.98802) yields a better chi-square fit to the classical sex-ratio data than both the binomial and beta-binomial, though the model is still rejected by a formal goodness-of-fit test.
The construction is less flexible than the beta-binomial—it cannot produce U-shaped pmfs—so it is most appropriate for data that look binomial apart from extra tail mass.

Where Pith is reading between the lines

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

The same construction works for any two discrete pmfs on the same support that share a mean; the bi-uniform is just the simplest block-shaped companion, so the recipe could be re-run with other tail-heavy companions.
The variance formula has a breakpoint at floor(np), and the positivity proof splits on whether np is an integer, suggesting that the mixture's properties—and its fit—may jump when p crosses a value where np is an integer.
A natural testable extension is to replace the single breakpoint with a smoother two-parameter tail correction, trading the model's simplicity for wider applicability while keeping the mean fixed.
The paper's caution about diagnosing overdispersion—raw variance comparison almost always shows a difference—directly supports using dispersion tests and effect sizes rather than point estimates when deciding whether a heavier-tailed model is warranted.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A small but honest paper: a new two-parameter mixture that trades flexibility for simplicity, with correct math and a transparent application.

read the letter

Short version: this is a modest but sound paper. The bi-uniform distribution is new, and the mixture with the binomial is a clean way to add tail weight while keeping the mean fixed. The math checks out; the variance decomposition is correct, and the positivity proof for the variance difference is fine. The Geissler application is transparent, including the fact that every fitted model is rejected by the chi-square test and that effect sizes are small.

What the paper does well: it states its own scope honestly. The mixture is not claimed to be a universal overdispersion fix; it is offered as a simple alternative that can beat the beta-binomial on some data. The derivation of the variance formula is straightforward and verified numerically. The exposition is clear enough to be used in an undergraduate statistics course, which seems to be the author's intent.

Soft spots, in proportion: the heavy-tail property is largely by construction — the bi-uniform was chosen to put more weight in the tails, so the mixture's improved tail fit is not a surprising empirical discovery. The weight w is chosen by minimum chi-square via a grid search on the same data, so the reported chi-square is optimistically biased; the paper does not report standard errors or any uncertainty on the parameter estimates. Also, the beta-binomial comparison is not fully apples-to-apples: beta-binomial uses method of moments while the mixture uses minimum chi-square for w. These are real but minor issues given the paper's narrow claims. The paper does not over-sell the result; it notes that other data may favor the beta-binomial.

One small point: the paper's own goodness-of-fit test rejects the mixture too, so the improvement in chi-square is a matter of degree, not a decisive win. The effect size measure, Cohen's w, is below 0.1 for all models, which the paper acknowledges.

Who this is for: people who teach discrete distributions or work on overdispersed count data and want a simple, derivable alternative to the beta-binomial. It won't reshape practice, but it is a legitimate small contribution. The citations are appropriate; the paper does not hide the existence of beta-binomial or its limitations.

Recommendation: send it to peer review. It deserves a serious referee; the math is correct and the writing is honest. The referee will likely ask for standard errors and a note about the optimism in the min-chi-square fit, but those are fixable. I would not cite it in my own work in the next year, but I would consider assigning it in a grad course as an example of building and checking a mixture model.

Referee Report

0 major / 6 minor

Summary. The paper proposes a new discrete distribution on {0,...,n}, formed as a weighted arithmetic mean of a binomial pmf and a newly introduced 'bi-uniform' pmf with the same mean np. It derives the bi-uniform's pmf and variance, proves that its variance exceeds the binomial variance for 0<p<1 and n>1, and develops method-of-moments and minimum-chi-square estimation for the mixing weight w. The model is applied to Geissler's classical sex-ratio data (n=8, m=53680), where the mixture achieves a Pearson chi-square of 43.58 versus 91.87 for the binomial and 53.31 for the beta-binomial, while preserving the same fitted mean. The paper is explicitly pedagogical in tone and positions the construction as a simple, tractable alternative to the beta-binomial for overdispersed count data.

Significance. If the result holds, the paper offers a genuinely elementary way to model extra-binomial variance: the mixture is defined by a weighted arithmetic mean of two pmfs, requires no calculus or continuous compounding, and preserves the binomial mean by construction. The variance formula and the positivity proof are correct and self-contained, and the numerical application reproduces the reported chi-square values. The contribution is modest but useful for teaching and for simple data fitting. The paper is honest about its limitations: it does not claim the bi-uniform component is a realistic data-generating mechanism, and it notes that the more flexible beta-binomial will be preferable in other settings. The availability of a closed-form variance and a simple moment estimator are concrete strengths.

minor comments (6)

[Section 3] In the derivation of the bi-uniform expectation, the second displayed equality has a sign error: the term involving p2 should be added, not subtracted, after expanding the sum over {floor(mu)+1,...,n}. The following system and the final formulas p1 and p2 are correct, but the intermediate step as printed is inconsistent and will confuse readers.
[Section 5.3 / Table 2] The minimum-chi-square estimate of w is obtained by a dense grid search on the same data that are later used to evaluate the fit. This is in-sample optimization and may overstate the improvement. The paper partially addresses this by also reporting the moment-based mixture ('Mixture (mom)'), whose chi-square is 45.47, still below the beta-binomial's 53.31. Still, the comparison of the grid-search mixture with the moment-fitted beta-binomial is not apples-to-apples; a sentence acknowledging this and, ideally, a beta-binomial minimum-chi-square fit or a small bootstrap for w would strengthen the claim.
[Section 5.3] No standard errors or confidence intervals are reported for the fitted parameters p and w. Given the very large sample size (m=53680), the uncertainty is likely small, but a brief bootstrap or delta-method statement would make the application more complete and would help readers gauge the stability of the chi-square comparison.
[Section 5.4] The sentence explaining degrees of freedom reads 'n is minus one the number of categories', which is garbled. The intended meaning is that there are n+1 categories, so df = n - k after one degree is lost to the total. Please rephrase for clarity.
[Table 1 caption] Typo: 'observ counts' should be 'observed counts'.
[Section 4.1.1] The condition for 0 <= w <= 1 is written with potentially ambiguous notation. It should clearly state: hat(v)_1 <= s^2 <= hat(v)_2, where hat(v)_1 and hat(v)_2 are the variance estimates obtained by replacing p with hat(p). The current formatting makes it look like squared quantities.

Circularity Check

0 steps flagged

No significant circularity: the heavy-tail construction is explicit, the derivations are parameter-free, and the application is a transparent in-sample fit.

full rationale

The paper's central claims are the construction of a bi-uniform distribution, the derivation of its variance, the proof that its variance exceeds the binomial variance, and the resulting mixture. None of these reduces to its inputs by construction in a misleading way. The bi-uniform is defined by two probability values and the requirement that its expectation equal a specified value μ; the fact that the mixture then has expectation np is openly a design constraint, not a discovered prediction. The variance increase is a genuine algebraic theorem: v2 is derived from the definition, and d(n,p)=v2-v1 is proved positive for all valid n,p without fitting any data. The paper also transparently states that the bi-uniform was introduced 'not with the intention of proposing a distribution for actual applications, but for providing a tool useful in putting more weight on the tails.' The application to Geissler's data is a fitting exercise, not an out-of-sample prediction. The minimum-chi-square estimate of w guarantees that the fitted mixture has chi-square no larger than the nested binomial (w=1), but this is explicitly presented as estimation, not as a forecast; moreover the moment-based mixture, whose w is not chosen to minimize chi-square, also yields a smaller chi-square (45.47) than the binomial (91.87), providing independent in-sample evidence. No load-bearing self-citations appear: all references (Fisher, Skellam, Johnson et al., Hogg et al., etc.) are external. The paper also acknowledges the mixture's limited flexibility and that other data circumstances may favor the beta-binomial. The reported numerical expected frequencies and chi-square values are consistent with the formulas. Therefore there is no circular derivation chain or disguised fitted-input-as-prediction.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 1 invented entities

The central model has only two fitted parameters (p and w); the bi-uniform distribution is an invented mathematical construct with no independent evidence. The main mathematical derivations are self-contained and correct.

free parameters (2)

p = 0.51468
Binomial success probability estimated by method of moments as sample mean / n (Section 5.1).
w = 0.98802 (min chi-square) / 0.98526 (method of moments)
Mixing weight estimated to match variance or minimize chi-square on the Geissler data (Sections 4.1 and 5.3).

axioms (3)

standard math Binomial distribution formulas and properties (pmf, expectation np, variance np(1-p))
Used throughout Sections 1 and 4 as the basis of the mixture.
standard math Pearson chi-square statistic follows a chi-square distribution with n-k degrees of freedom under the null hypothesis
Used for goodness-of-fit tests in Section 5.4.
domain assumption Geissler's 53,680 families are an independent and identically distributed sample with constant p
Section 5.1; essential for treating the frequency table as binomial/mixture data and for the chi-square approximation.

invented entities (1)

bi-uniform distribution no independent evidence
purpose: Acts as the second component in a mixture with the binomial to shift probability mass from the center to the tails while preserving the mean.
Introduced in Section 3 as a mathematical definition; it has no external falsifiable handle, being a tool for constructing the mixture rather than an empirically motivated entity.

pith-pipeline@v1.3.0-alltime-deepseek · 7918 in / 11285 out tokens · 500583 ms · 2026-08-01T16:08:33.653934+00:00 · methodology

0 comments

read the original abstract

A simple alternative to the binomial distribution that places more probability weight on the tails is considered. Its derivation only requires the weighted arithmetic mean of two discrete probability mass functions, one being the binomial itself and the other being the bi-uniform introduced here. Some properties are derived, and an application to a classical data set is discussed. The presentation can be seen as an exemplary treatise on how to construct a statistical model, derive statistical properties, fit models to actual data by employing estimation methods, and verify appropriateness by using elements from statistical hypothesis testing.

Figures

Figures reproduced from arXiv: 2607.18083 by J\"urgen Gro{\ss}.

**Figure 1.** Figure 1: Variance estimates vb1 and s 2 for 1000 samples of size m = 100 from a binomial distribution with parameters n = 17 and p = 0.4. close to either 0 or n. The variance of the binomial distribution is v1 := Var(X) = E(X2 ) − [E(X))]2 = np(1 − p) . See also Section 3.1 in Hogg et al. (2019) for derivations and further properties of the binomial distribution. When there are m observations x1, . . . , xm repres… view at source ↗

**Figure 2.** Figure 2: Graphical representation of probability mass functions for the case n = 17 (upper panel) and their differences (lower panel). 1.2. Alternative Distributions. An often proposed alternative when empirical frequencies do not resemble the binomial probabilities is the so-called betabinomial distribution. This distribution is briefly introduced in the following Section 2. Then, as the main objective of this … view at source ↗

**Figure 3.** Figure 3: Binomial pmf for the case n = 17 and p = 0.4 compared with bi-uniform pmf with µ = np. Assume now that a very similar pmf is sought with the property that its expectation is not necessarily in the center but may take any specified value between 0 and n. A successful approach allows the usage of only two different probability values instead of 1/(n+ 1). To see this, let µ be any real number with 0 ≤ µ < n a… view at source ↗

**Figure 4.** Figure 4: Variance of the bi-uniform pmf minus variance of the binomial pmf, i.e. d(n = 17, p) as a function of 0 < p < 1. µ = np. If we let w be some weight factor such that 0 ≤ w ≤ 1, then the mixture pX(x) = w · p (1) X (x) + (1 − w) · p (2) X (x) is again a pmf on M. Moreover, it has expectation E(X) = np and variance Var(X) = w · v1 + (1 − w) · v2 . In general, the variance of a mixture is not the weighted ave… view at source ↗

**Figure 5.** Figure 5: Contributions to the chi-square statistic from three fitted models. mostly lacks fit in the center but slightly also for the extreme values 0 and 8. 5.3. Mixture Model. In Tables 1 & 2 and in [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

discussion (0)

Reference graph

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Abramowitz, M. and Stegun, I. (1972),Handbook of Mathematical Functions, National Bureau of Stan- dards, Applied Mathematics Series

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V., McKean, J

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Ishii, G. and Hayakawa, R. (1960), ‘On the compound binomial distribution’,Annals of the Institute of Sta- tistical Mathematics12(1), 69–80

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