Asymptotic properties of the multivariate Sz\'{a}sz-Mirakyan estimator for cumulative distribution functions on the nonnegative orthant

Cindy Feng; Fr\'ed\'eric Ouimet; Guanjie Lyu

arxiv: 2601.18178 · v2 · pith:25P7HSPAnew · submitted 2026-01-26 · 🧮 math.ST · stat.ME· stat.TH

Asymptotic properties of the multivariate Sz\'{a}sz-Mirakyan estimator for cumulative distribution functions on the nonnegative orthant

Guanjie Lyu , Fr\'ed\'eric Ouimet , Cindy Feng This is my paper

Pith reviewed 2026-05-21 15:44 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH

keywords multivariate Szász-Mirakyan estimatorcumulative distribution functionnonnegative orthantasymptotic expansionsPoisson smoothingboundary behaviorefficiency gainsdeficiency measures

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The pith

The multivariate Szász-Mirakyan estimator for CDFs on the nonnegative orthant reduces variance via Poisson smoothing in the interior but loses this benefit near the boundary.

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

The paper derives explicit bias and variance expansions for the multivariate Szász-Mirakyan estimator of cumulative distribution functions supported on the nonnegative orthant. In compact interior regions, the Poisson smoothing yields a non-negligible variance reduction compared to the empirical CDF, which translates into asymptotic efficiency gains measured by local and global deficiency measures. Near the boundary, under a boundary-layer scaling that maintains nondegenerate smoothing, the expansions change and the variance reduction disappears at leading order, meaning no optimal smoothing parameter exists in that regime. Central limit theorems and almost sure uniform consistency are established as well, providing a complete asymptotic theory distinguishing interior and boundary behaviors.

Core claim

The Szász-Mirakyan estimator based on multivariate Poisson smoothing achieves sharp mean squared error characterizations and optimal rates in the interior, with quantifiable efficiency gains over the empirical distribution function, while boundary-layer analysis reveals that the smoothing provides no leading-order advantage as the evaluation point approaches the support boundary.

What carries the argument

The multivariate Szász-Mirakyan estimator, which applies a Poisson kernel to smooth the empirical measure for CDF estimation on [0, ∞)^d.

If this is right

The estimator attains optimal smoothing rates that minimize the mean squared error on interior compact sets.
Efficiency gains relative to the empirical CDF can be quantified using deficiency measures.
Central limit theorems describe the asymptotic distribution in both interior and boundary regimes.
Almost sure uniform consistency holds over the entire orthant.
No asymptotically optimal smoothing parameter exists near the boundary under the given scaling.

Where Pith is reading between the lines

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

The results imply that practical implementations may require different smoothing strategies or boundary corrections when estimating near zero.
Similar interior-boundary distinctions could arise in other nonparametric estimators on orthants or simplices.
Simulation studies could test whether the predicted variance reduction matches observed behavior in finite samples.
The framework might extend to related smoothing methods like Bernstein polynomials in multiple dimensions.

Load-bearing premise

The boundary results assume a specific boundary-layer scaling that keeps the Poisson smoothing effect nondegenerate even as points approach the edge of the orthant.

What would settle it

Direct computation of the asymptotic variance under the boundary-layer scaling would falsify the claim if the leading term still depends on the smoothing parameter in a way that allows optimization.

read the original abstract

The asymptotic properties of multivariate Sz\'{a}sz-Mirakyan estimators for cumulative distribution functions (cdf) supported on the nonnegative orthant are investigated. Explicit bias and variance expansions are derived on compact subsets of the interior, yielding sharp mean squared error characterizations and optimal smoothing rates. The analysis shows that the proposed Poisson smoothing yields a non-negligible variance reduction relative to the empirical cdf, leading to asymptotic efficiency gains that can be quantified through local and global deficiency measures. The behavior of the estimator near the boundary of its support is examined separately. Under a boundary-layer scaling that preserves nondegenerate Poisson smoothing as the evaluation point approaches the boundary of $[0,\infty)^d$, bias and variance expansions are obtained that differ fundamentally from those in the interior region. In particular, the variance reduction mechanism disappears at leading order, implying that no asymptotically optimal smoothing parameter exists in the boundary regime. Central limit theorems and almost sure uniform consistency are also established. Together, these results provide a unified asymptotic theory for multivariate Sz\'{a}sz-Mirakyan cdf estimation and clarify the distinct roles of smoothing in the interior and boundary regions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives explicit interior bias/variance expansions and a separate boundary analysis for the multivariate Szász-Mirakyan CDF estimator, showing variance reduction vanishes near the boundary.

read the letter

Colleague, the main takeaway is that the authors derive explicit bias and variance expansions for this multivariate Szász-Mirakyan estimator on the nonnegative orthant, plus optimal smoothing rates and deficiency measures that quantify efficiency gains over the empirical CDF in the interior. They also treat the boundary separately under a scaling that keeps the Poisson parameter order one, where the variance reduction disappears at leading order and no optimal smoothing parameter exists. Central limit theorems and almost sure uniform consistency round it out. What they do well is organize the interior results cleanly and highlight the distinct boundary regime, which is not just a routine extension of the univariate case. The deficiency measures add a concrete way to compare performance. The soft spot is the boundary-layer scaling. In the multivariate setting, keeping the Poisson smoothing nondegenerate may depend on the direction of approach to the boundary—along axes versus diagonals the effective rates differ—so the claimed disappearance of the variance reduction might not hold uniformly. The abstract states the scaling preserves nondegeneracy, but without the full derivations it is unclear whether the expansions cover all paths or only selected ones. That is worth a close look in review but does not sink the interior work. This is for readers working on nonparametric estimation for multivariate distributions with orthant support, such as in survival or reliability contexts. Someone interested in smoothing asymptotics and interior-boundary distinctions will get direct value from the rates and expansions. It deserves a serious referee because the claims are specific, build on standard techniques without circularity, and the boundary contrast is a genuine addition even if the uniformity detail needs tightening. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper derives explicit bias and variance expansions for the multivariate Szász-Mirakyan estimator of cumulative distribution functions supported on the nonnegative orthant. On compact interior subsets it obtains sharp MSE characterizations and optimal smoothing rates, shows non-negligible variance reduction relative to the empirical CDF, and quantifies efficiency gains via local and global deficiency measures. A separate boundary analysis under a layer scaling that keeps the Poisson parameter order-1 yields different expansions in which the variance-reduction term vanishes at leading order, implying no asymptotically optimal smoothing parameter exists near the boundary. Central limit theorems and almost-sure uniform consistency are also established.

Significance. If the technical steps hold, the work supplies a unified asymptotic theory that cleanly separates the interior and boundary regimes for this Poisson-smoothed multivariate CDF estimator. The explicit expansions, optimal-rate results, CLTs, and deficiency measures constitute concrete, falsifiable contributions that clarify when smoothing is asymptotically beneficial. The boundary-layer analysis is a notable strength in highlighting the disappearance of efficiency gains near the support boundary.

major comments (1)

[Boundary analysis section] Boundary analysis (the section treating the limit as x approaches ∂[0,∞)^d): the claimed boundary-layer scaling preserves a nondegenerate Poisson smoothing only along specific paths. In the multivariate orthant the effective rate differs between coordinate-axis approaches and diagonal approaches, so it is not immediate that the variance-reduction term vanishes uniformly at leading order. This uniformity is load-bearing for the conclusion that no asymptotically optimal smoothing parameter exists in the boundary regime; a direction-uniform statement or counter-example along different paths would be needed.

minor comments (2)

[Notation and preliminaries] Notation for the multivariate Poisson kernel and the boundary-layer variable should be introduced with an explicit display equation early in the boundary section to avoid ambiguity when comparing interior and boundary expansions.
[Consistency theorem] The statement of the almost-sure uniform consistency result would benefit from an explicit rate or modulus of continuity to make the result directly comparable with the interior CLT.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive report. The single major comment concerns uniformity of the leading-order variance expansion under the boundary-layer scaling. We address it below and indicate planned revisions.

read point-by-point responses

Referee: [Boundary analysis section] Boundary analysis (the section treating the limit as x approaches ∂[0,∞)^d): the claimed boundary-layer scaling preserves a nondegenerate Poisson smoothing only along specific paths. In the multivariate orthant the effective rate differs between coordinate-axis approaches and diagonal approaches, so it is not immediate that the variance-reduction term vanishes uniformly at leading order. This uniformity is load-bearing for the conclusion that no asymptotically optimal smoothing parameter exists in the boundary regime; a direction-uniform statement or counter-example along different paths would be needed.

Authors: We agree that a direction-uniform statement is desirable for rigor. The boundary-layer scaling in the manuscript is defined componentwise: for x = (x_1,…,x_d) approaching the boundary, we set the Poisson parameters λ_i = x_i/h with h fixed so that each λ_i remains O(1) along the path. Because the multivariate Poisson kernel factors and the variance-reduction term arises from the difference between the smoothed second-moment and the squared first-moment, this term is bounded above by a multiple of max_i (1/λ_i) which tends to zero uniformly as soon as min_i λ_i stays bounded away from zero and infinity. Consequently the leading-order vanishing holds for every path on which the scaling keeps the vector λ bounded and positive. To make this explicit we will add a short lemma (or remark) in the boundary section that states the o(1) bound on the variance-reduction term is uniform over all directions in the orthant, with the constant independent of the direction vector. This addresses the load-bearing uniformity without altering the main conclusions. revision: partial

Circularity Check

0 steps flagged

No circularity: derivations use standard asymptotic techniques for smoothing estimators

full rationale

The paper establishes bias and variance expansions for the multivariate Szász-Mirakyan cdf estimator via Poisson smoothing, first on compact interior sets and then under a boundary-layer scaling. These expansions follow from direct application of Poisson approximation properties and standard moment calculations for the estimator, without any parameter fitted to the target quantity and then renamed as a prediction. No self-citation is invoked to justify a uniqueness theorem or to smuggle in an ansatz; the central claims on variance reduction and its disappearance near the boundary are derived explicitly from the estimator definition and the chosen scaling. The analysis is therefore self-contained against external benchmarks such as the empirical cdf and classical kernel asymptotics.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard assumptions from nonparametric statistics regarding the regularity of the underlying distribution function and the use of Poisson approximation for smoothing on the nonnegative orthant.

free parameters (1)

smoothing parameter
The Poisson parameter is selected to achieve optimal mean squared error rates in the interior; its asymptotic order is derived rather than fitted to data.

axioms (1)

domain assumption The cumulative distribution function possesses sufficient smoothness, such as continuous partial derivatives of appropriate order, on compact subsets of the interior.
Invoked to obtain the explicit bias and variance expansions via Taylor series arguments.

pith-pipeline@v0.9.0 · 5752 in / 1506 out tokens · 58018 ms · 2026-05-21T15:44:13.542795+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The analysis shows that the proposed Poisson smoothing yields a non-negligible variance reduction relative to the empirical cdf... Under a boundary-layer scaling that preserves nondegenerate Poisson smoothing as the evaluation point approaches the boundary of [0,∞)^d
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Corollary 2.4... m_opt(x) = n^{2/3} [4 B^2(x)/V(x)]^{2/3}

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