Equal probabilities maximize the expected deficit in the siblings of the coupon collector
Pith reviewed 2026-06-26 13:17 UTC · model grok-4.3
The pith
Equal probabilities for coupon types strictly maximize the expected missing coupons for each sibling collector.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For every fixed N and every j ≥ 2, E[U_j^N] is strictly larger at the uniform probability vector than at any other vector, and it strictly increases along every ray from an arbitrary distribution toward the uniform one. The argument extends without change to all real j > 1. By-products include a finite closed form for E[U_j^N] over subsets of the coupon set and the exact Hessian at the uniform vector.
What carries the argument
The radial derivative of the Poissonized expectation, obtained after inclusion-exclusion and one integration by parts, rewritten as a positively weighted covariance of an increasing function whose sign is settled by Chebyshev's correlation inequality.
Load-bearing premise
The Poissonized integral representation of the expectation admits an exact radial derivative that reduces, after one integration by parts, to a positively weighted covariance of an increasing function.
What would settle it
Direct numerical evaluation of E[U_j^N] for small N such as 3 and j=2 at two points along any chosen ray from a non-uniform distribution to the uniform vector, checking whether the values strictly increase.
read the original abstract
In the siblings (or brotherhood) variant of the coupon collector's problem, a main collector draws coupons until her own album is complete and passes every duplicate down a chain of siblings; the $j$th collector is then left with $U_j^N$ empty places, $j\ge 2$. It has been conjectured [stated as an open problem in the work that introduced the model] that, for every fixed number of coupon types $N$ and every $j\ge 2$, the expected deficit $\E[U_j^N]$ is maximized by the equiprobable coupon distribution. We prove this in a sharp, finite-$N$ form: $\E[U_j^N]$ is strictly larger at the uniform vector than at any other probability vector, and indeed strictly increases along every ray running from an arbitrary distribution toward the uniform one. The proof is exact and elementary in its ingredients. An inclusion--exclusion step turns the governing Poissonized integral into a one-dimensional integral with a separable integrand; a single integration by parts then rewrites the radial derivative of $\E[U_j^N]$ as a positively weighted covariance of an increasing function, whose sign is settled by Chebyshev's correlation inequality. We show that $\E[U_j^N]$ is \emph{not} Schur-concave, so that no majorization or pairwise-smoothing argument can yield the result, and we explain why the recent variance-extremality method of Long~[Long, arXiv:2604.25108, 2026] does not transfer. As by-products we obtain a finite closed form for $\E[U_j^N]$ over subsets of the coupon set and the exact Hessian of $\E[U_j^N]$ at the uniform vector. The argument extends without change to all real $j>1$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that in the siblings variant of the coupon collector problem, the expected deficit E[U_j^N] for the j-th collector (j >= 2) is strictly maximized at the uniform probability vector over N types, and strictly increases along every ray from an arbitrary distribution toward uniformity. The proof converts the Poissonized expectation via inclusion-exclusion to a one-dimensional integral with separable integrand, applies one integration by parts to rewrite the radial derivative as a positively weighted covariance of an increasing function, and invokes Chebyshev's correlation inequality for the sign. By-products include an explicit closed form for E[U_j^N] on subsets and the exact Hessian at uniformity. The result extends to real j > 1; the paper also shows E[U_j^N] is not Schur-concave (precluding majorization arguments) and explains why Long's variance-extremality method does not apply.
Significance. If the result holds, it resolves an open conjecture with a sharp, finite-N analytic proof that relies only on elementary steps (inclusion-exclusion, Poissonization, integration by parts, Chebyshev) and supplies parameter-free closed forms plus the Hessian. These explicit constructions and the direct ray-monotonicity argument (avoiding Schur-concavity) constitute a substantive contribution to the literature on coupon-collector variants and extremal inequalities for expectations in stochastic processes.
minor comments (1)
- In the paragraph outlining the proof ingredients, the transition from the separable integrand to the covariance expression after integration by parts would be clearer if the resulting weighted-covariance formula were displayed explicitly as an equation.
Simulated Author's Rebuttal
We thank the referee for the careful reading of the manuscript and for the positive evaluation. The recommendation to accept is appreciated, and we are pleased that the elementary nature of the proof and the explicit constructions were viewed as substantive contributions.
Circularity Check
No significant circularity
full rationale
The central claim is proved by converting the Poissonized expectation to a one-dimensional integral via inclusion-exclusion, followed by a single integration by parts that expresses the radial derivative as a weighted covariance, whose sign is fixed by Chebyshev's inequality. These steps are direct applications of standard analytic tools and do not reduce the target quantity E[U_j^N] to any fitted parameter, self-defined input, or self-citation chain. The paper explicitly notes that it is not Schur-concave and that a prior variance method does not transfer, but invokes no load-bearing uniqueness theorem or ansatz from prior work by the same authors. The derivation is therefore self-contained against external mathematical facts.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Chebyshev's correlation inequality applies to the covariance obtained after integration by parts
- domain assumption The Poissonized expectation equals the original discrete expectation for the deficit functional
Reference graph
Works this paper leans on
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[1]
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[2]
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Foata and D
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[7]
C. D. Long, Terminal defects, growing multiplicity, and variance extremality in the double Dixie cup problem, preprint (2026), arXiv:2604.25108
Pith/arXiv arXiv 2026
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[10]
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discussion (0)
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