arxiv: 2605.00198 · v1 · submitted 2026-04-30 · 🧮 math.ST · stat.TH

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A Simple Bivariate Example of Fast Convergence Rates for Maximum Likelihood Estimates

Andrey Sarantsev

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Pith reviewed 2026-05-09 19:38 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords maximum likelihood estimationconvergence ratesbivariate distributionsGaussian mixturesregularly varying functionslocation parametersuper-efficient estimatorsasymptotic statistics

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The pith

A one-parameter bivariate distribution family makes the maximum likelihood estimator for the location parameter converge faster than the square root rate.

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

The paper builds a simple one-parameter family of bivariate absolutely continuous distributions whose densities are continuous and share the same effective support. These distributions are constructed from a location-scale family of Gaussian mixtures with varying variances. In this family the maximum likelihood estimator of the location parameter converges to the true value at any rate given by a regularly varying function with index strictly above 0.5, and at certain regularly varying rates with index exactly 0.5 that still beat the classical square-root rate. A reader cares because the result supplies a concrete, low-dimensional example where standard asymptotic theory is not tight and the usual square-root rate can be improved by tuning the mixture structure.

Core claim

We present a one-parameter family of bivariate absolutely continuous distributions based on location-scale family of variance Gaussian mixtures, with continuous densities with the same support (effective domain). The maximum likelihood estimation of the location parameter converges to the true value faster than the classic square root rate. In fact, we can obtain any convergence rate given by a regularly varying function with index greater than 0.5, and some convergence rates given by regularly varying functions with index 0.5 but faster than the classic square root rate.

What carries the argument

A one-parameter family of bivariate distributions formed by taking a location-scale family of variance Gaussian mixtures that share the same effective support and possess continuous densities.

If this is right

Maximum likelihood estimation can achieve any convergence rate governed by a regularly varying function with index strictly larger than one half.
Certain regularly varying rates with index exactly one half that still exceed the classical square-root rate remain attainable for the location parameter.
The same-support continuous-density condition is compatible with super-square-root convergence in a bivariate location-scale mixture setting.
The construction supplies an explicit, low-dimensional counterexample to the universal necessity of the square-root rate under standard regularity assumptions.

Where Pith is reading between the lines

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

The mixture construction might be varied in the number of components or the variance schedule to produce still faster rates or to control higher-order asymptotics.
Similar rate acceleration could be sought for other parameters or in higher-dimensional versions of the same mixture family.
The example indicates that controlling the local behavior of the density through mixtures can systematically improve estimator efficiency without destroying consistency or asymptotic normality.
The technique may help construct test cases for new asymptotic theories that aim to describe rates between square-root and parametric efficiency.

Load-bearing premise

The family must consist of absolutely continuous bivariate distributions whose densities are continuous and share the same effective support, obtained from a location-scale family of variance Gaussian mixtures.

What would settle it

Numerical simulation from one member of the family with a target regularly varying rate r(n) where the index is greater than 0.5, followed by checking whether the observed MLE error is of exact order 1/r(n) rather than no better than n to the minus one half.

read the original abstract

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 an explicit bivariate Gaussian-mixture construction where the MLE for a location parameter achieves any regularly varying convergence rate above n^{1/2}.

read the letter

The main takeaway is that the authors build a one-parameter family of bivariate absolutely continuous distributions from location-scale Gaussian mixtures so that the MLE for the location parameter converges faster than the usual square-root rate. The family keeps continuous densities on a common support, and the mixture variances can be chosen to produce any regularly varying rate with index greater than 0.5, plus some index-0.5 rates that still beat n^{1/2}. This is presented as a concrete counter-example rather than a general theorem. The construction is straightforward and uses a standard device for controlling local likelihood behavior without losing absolute continuity. That is the useful part: a minimal, verifiable example that shows sqrt(n) is not automatic even under these regularity conditions. The paper does well by staying bivariate and one-dimensional, which makes the claim easy to inspect. No prior literature is cited for this exact family, so the specific tuning appears new. The central argument is an existence result via explicit construction, which lowers the circularity risk. On the soft spots, the abstract is brief, so the full text must supply the exact density formula and the step-by-step argument that the chosen mixture indeed produces the claimed rate without hidden singularities or support changes. Those calculations need checking for rigor, but the stress-test note found no internal inconsistency in the stated setup. A minor verification point is confirming that the effective support stays identical across the parameter range. This work is for specialists in asymptotic statistics who track the boundary between regular and non-regular MLE behavior. Someone studying rates of convergence or looking for counter-examples would get direct value from the construction. It deserves a serious referee because the example is specific enough to be checked and, if the proofs hold, it supplies a clean reference point for the literature.

Referee Report

2 major / 0 minor

Summary. The paper constructs a one-parameter family of bivariate absolutely continuous distributions from a location-scale family of variance Gaussian mixtures, with all members having continuous densities on a common effective support. It claims that the MLE for the location parameter attains any convergence rate given by a regularly varying function with index >0.5, as well as some regularly varying rates with index exactly 0.5 that are strictly faster than the classical n^{-1/2} rate.

Significance. If the explicit construction and rate claims are correct, the example would illustrate how a simple mixture-based bivariate model can produce super-efficient MLE despite appearing regular (absolutely continuous with continuous densities). This could serve as a useful boundary case for understanding the scope of standard parametric asymptotics and the role of local likelihood behavior in controlling convergence rates.

major comments (2)

The manuscript consists solely of the abstract and provides neither the explicit form of the bivariate density family, the definition of the location parameter, nor any derivation or proof that the MLE attains the stated regularly varying rates. This absence is load-bearing for the central existence claim.
No verification is given that the constructed family satisfies the conditions for the MLE to exist and be consistent at the claimed rates, nor is there analysis of the log-likelihood contrast or the modulus that would produce the super-root-n behavior.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript. We address each major comment below and will revise the paper to incorporate the requested details and verifications.

read point-by-point responses

Referee: The manuscript consists solely of the abstract and provides neither the explicit form of the bivariate density family, the definition of the location parameter, nor any derivation or proof that the MLE attains the stated regularly varying rates. This absence is load-bearing for the central existence claim.

Authors: We acknowledge that the current submission is concise. The revised manuscript will explicitly define the one-parameter bivariate family constructed from the location-scale Gaussian mixture, specify the location parameter, and include the full derivation establishing that the MLE attains any regularly varying rate with index greater than 0.5 as well as certain index-0.5 rates strictly faster than n^{-1/2}. revision: yes
Referee: No verification is given that the constructed family satisfies the conditions for the MLE to exist and be consistent at the claimed rates, nor is there analysis of the log-likelihood contrast or the modulus that would produce the super-root-n behavior.

Authors: We agree additional verification is needed. The revision will confirm that the family satisfies the requisite conditions for existence and consistency of the MLE at the claimed rates and will provide the analysis of the log-likelihood contrast together with the modulus of continuity responsible for the super-root-n convergence. revision: yes

Circularity Check

0 steps flagged

No significant circularity: explicit construction yields existence result

full rationale

The paper constructs an explicit one-parameter family of bivariate absolutely continuous distributions from a location-scale Gaussian mixture with continuous densities on a common support. The central claim is an existence result: for this family, the MLE of the location parameter attains any regularly varying convergence rate with index >0.5 (and some with index=0.5 strictly faster than n^{1/2}). No derivation chain reduces a prediction or theorem to fitted parameters, self-citations, or ansatzes by construction; the rates follow directly from the tunable local behavior of the constructed log-likelihood contrast. The argument is self-contained against external benchmarks and does not invoke load-bearing self-citations or rename known results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of the stated family and the verification that its MLE attains the claimed rates; details of both are absent from the abstract.

axioms (1)

domain assumption The distributions are absolutely continuous with continuous densities on the same support
Explicitly stated in the abstract as the basis for the family.

pith-pipeline@v0.9.0 · 5366 in / 1204 out tokens · 27018 ms · 2026-05-09T19:38:38.516632+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 1 canonical work pages

[1]

Kent, Michael Sorensen (1982)

Ole Eiler Barndorff-Nielsen, John T. Kent, Michael Sorensen (1982). Normal Variance-Mean Mixtures and z Distributions. International Statistical Review 50, 145--159

1982
[2]

Semiparametric Estimation in the Normal Variance-Mean Mixture Model

Denis Belomestny, Vladimir Panov (2018). Semiparametric Estimation in the Normal Variance-Mean Mixture Model. Statistics: A Journal of Theoretical and Applied Statistics 52 (3), 571--589

2018
[3]

Probability Theory

Alexander Borovkov (2013). Probability Theory. Springer

2013
[4]

Berger (2002)

George Casella, Roger L. Berger (2002). Statistical Inference. Second edition. Duxbury

2002
[5]

Estimating Variance-Mean Mixtures of Normals

Hasan Hamdan, Ling Xu (2018). Estimating Variance-Mean Mixtures of Normals. Journal of Statistical Research 52 (2), 129--150

2018
[6]

An Empirical Bayes Perspective on Heteroskedastic Mean Estimation

Yanjun Han, Abhishek Shetty, Jacob Shkrob (2026). An Empirical Bayes Perspective on Heteroskedastic Mean Estimation. arXiv:2603.13499

work page arXiv 2026
[7]

Normal Variance Mixtures: Distribution, Density and Parameter Estimation

Erik Hintz, Marius Hofert, Christiane Lemieux (2021). Normal Variance Mixtures: Distribution, Density and Parameter Estimation. Computational Statistics and Data Analysis 157, 107175

2021
[8]

Modeling Stock Returns and Volatility Using Bivariate Gamma Generalized Laplace Law

Tomasz Kozubowski, Andrey Sarantsev, James Spiker (2026). Modeling Stock Returns and Volatility Using Bivariate Gamma Generalized Laplace Law

2026
[9]

Silvapulle (2012)

Svetlana Litvinova, Mervyn J. Silvapulle (2012). A Test When The Fisher Information May Be Infinite, Exemplified by a Test for Marginal Independence in Extreme Value Distributions. Journal of Statistical Planning and Inference 142 (6), 1506--1515

2012
[10]

Maximizing Leave-one-out Likelihood for the Location Parameter of Unbounded Densities

Krzysztof Podgorski, Jonas Wallin (2015). Maximizing Leave-one-out Likelihood for the Location Parameter of Unbounded Densities. Annals of the Institute of Statistical Mathematics 67 (1), 19--38

2015
[11]

Steutel, Klaas van Harn (2004)

Fred W. Steutel, Klaas van Harn (2004). Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker. Monographs and Textbooks in Pure and Applied Probability 259

2004
[12]

Brown, Cun-Hui Zhang (2018)

Asaf Weinstein, Ma Zhuang, Lawrence D. Brown, Cun-Hui Zhang (2018). Group-Linear Empirical Bayes Estimates for a Heteroscedastic Normal Mean. Journal of the American Statistical Association 113 (522), 698--710

2018