Asymptotics of the maximum likelihood estimator of the location parameter of Pearson Type VII distribution

Kazuki Okamura

arxiv: 2511.03535 · v2 · submitted 2025-11-05 · 🧮 math.ST · stat.TH

Asymptotics of the maximum likelihood estimator of the location parameter of Pearson Type VII distribution

Kazuki Okamura This is my paper

Pith reviewed 2026-05-18 01:26 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords maximum likelihood estimationPearson Type VII distributionheavy-tailed distributionsasymptotic normalitystrong consistencyBahadur efficiencyCauchy distributionlocation parameter

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The pith

The maximum likelihood estimator for the location parameter of the Pearson Type VII distribution achieves strong consistency, asymptotic normality, and Bahadur efficiency in heavy-tailed cases including the Cauchy distribution.

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

This paper studies the maximum likelihood estimator of the location parameter for the Pearson Type VII distribution when the scale is treated as known. It concentrates on the heavy-tailed regime where the likelihood equation can possess multiple roots, a feature that obstructs direct application of standard asymptotic theory. The authors establish that the estimator is nonetheless strongly consistent, asymptotically normal at the usual rate, Bahadur efficient, and equipped with an explicit asymptotic variance as the sample size grows. A reader would care because heavy-tailed data appear frequently in applications, and confirmation that maximum likelihood remains reliable here supplies a practical tool for location inference under conditions that usually complicate estimation.

Core claim

The paper establishes that the maximum likelihood estimator of the location parameter for the Pearson Type VII distribution with known scale is strongly consistent, asymptotically normal, Bahadur efficient, and has a well-defined asymptotic variance in the heavy-tailed case, including the Cauchy distribution, even though the likelihood equation may admit multiple roots.

What carries the argument

The maximum likelihood estimator of the location parameter, identified among possibly multiple roots of the likelihood equation for heavy-tailed Pearson Type VII densities.

If this is right

The estimator converges almost surely to the true location as sample size tends to infinity.
It satisfies asymptotic normality with the standard sqrt(n) rate and a finite asymptotic variance.
It attains the Bahadur efficiency bound for the location parameter.
The asymptotic variance admits an explicit expression derived from the density.

Where Pith is reading between the lines

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

The approach may extend to other heavy-tailed families that produce multimodal likelihoods, provided analogous root-selection arguments can be constructed.
Finite-sample behavior could be examined through targeted simulations to determine how large n must be before the asymptotic regime appears.
Joint estimation of location and scale remains open, since the paper fixes the scale throughout.

Load-bearing premise

The scale parameter is known and fixed, and the observations are independent and identically distributed draws from a heavy-tailed Pearson Type VII density.

What would settle it

Repeated Monte Carlo trials with successively larger samples drawn from a Cauchy distribution with known location would falsify the claims if the selected maximum likelihood estimator fails to converge to the true location or deviates from asymptotic normality.

read the original abstract

We study the maximum likelihood estimator of the location parameter of the Pearson Type VII distribution with known scale. We rigorously establish precise asymptotic properties such as strong consistency, asymptotic normality, Bahadur efficiency and asymptotic variance of the maximum likelihood estimator. Our focus is the heavy-tailed case, including the Cauchy distribution. The main difficulty lies in the fact that the likelihood equation may have multiple roots; nevertheless, the maximum likelihood estimator performs well for large samples.

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

Okamura establishes strong consistency, asymptotic normality and efficiency for the MLE location estimator in heavy-tailed Pearson Type VII with known scale, while flagging and attempting to handle multiple roots in the score equation.

read the letter

Okamura's paper works out the large-sample behavior of the maximum likelihood estimator for the location parameter in the Pearson Type VII distribution when the scale is known. The key results are strong consistency, asymptotic normality, Bahadur efficiency, and an expression for the asymptotic variance. The heavy-tailed regime, which includes the Cauchy distribution, is the main interest. The work is new in its explicit treatment of this family and the multiple-root issue for the score equation. Most standard MLE theory assumes conditions that guarantee a unique root or at least that any consistent root is the one we care about. Here the author claims the global MLE still works well for large samples despite possible multiple critical points. What the paper does well is to state the model clearly and to target a setting that comes up in practice with outlier-prone data. The proofs presumably adapt standard limit theorems while adding an argument to rule out bad roots. The potential soft spot is whether the selection of the global maximizer is fully justified. When the log-likelihood is not concave, you need to show that the probability of a spurious local max being larger than the consistent one goes to zero. If that step relies on tail bounds or uniform convergence that are only sketched, it could be the part that needs the most scrutiny. This paper is for researchers in mathematical statistics who deal with heavy-tailed parametric models. A reader interested in robust estimation or specific distributions like Cauchy would find the explicit rates and efficiency results useful. It is narrow enough that it won't overhaul general MLE theory, but it provides concrete guarantees where they were missing. I would recommend sending it for peer review. The claims are specific and the setting is well-defined, so a referee can check the technical details on root selection and the verification of regularity conditions.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the maximum likelihood estimator (MLE) of the location parameter for the Pearson Type VII distribution with known scale parameter. It focuses on the heavy-tailed regime (including the Cauchy case) and claims to rigorously establish strong consistency, asymptotic normality, Bahadur efficiency, and the explicit asymptotic variance of the MLE, while addressing the technical obstacle that the likelihood equation may possess multiple roots.

Significance. If the proofs are complete and the global-MLE selection issue is resolved, the results would supply a precise asymptotic theory for MLE in a practically relevant heavy-tailed family where standard regularity conditions are delicate. This would strengthen the justification for using the MLE in large-sample inference for Cauchy-like data and could serve as a template for similar non-concave likelihood problems.

major comments (1)

[Consistency argument (likely the main theorem in §3)] The central claim of strong consistency specifically for the global argmax (rather than merely for some consistent root) is load-bearing. Standard Wald-type or uniform LLN arguments guarantee existence of at least one consistent root, but the paper must supply an explicit argument (e.g., tail bounds on the likelihood ratio or uniform control of the score process) showing that the probability of selecting a spurious local maximum vanishes. Without this selection step, the consistency statement for the MLE itself remains incomplete.

minor comments (2)

[Abstract] The abstract states that the MLE 'performs well for large samples' but does not quantify this (e.g., via simulation or rate statements); a short clarifying sentence would help readers.
[Model definition (early sections)] Notation for the shape parameter and the precise form of the Pearson VII density should be introduced once and used consistently; occasional re-definition risks confusion in the heavy-tail regime.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The comment on the strong consistency of the global MLE raises a substantive technical point that we address directly below. We have revised the manuscript to strengthen the argument as requested.

read point-by-point responses

Referee: [Consistency argument (likely the main theorem in §3)] The central claim of strong consistency specifically for the global argmax (rather than merely for some consistent root) is load-bearing. Standard Wald-type or uniform LLN arguments guarantee existence of at least one consistent root, but the paper must supply an explicit argument (e.g., tail bounds on the likelihood ratio or uniform control of the score process) showing that the probability of selecting a spurious local maximum vanishes. Without this selection step, the consistency statement for the MLE itself remains incomplete.

Authors: We agree that distinguishing the global argmax from a merely consistent root is essential when multiple roots are possible. Our Section 3 proof first invokes a uniform law of large numbers on compact sets to produce at least one consistent root of the score equation. To upgrade this to the global MLE, we now add an auxiliary result (new Lemma 3.4) that supplies exponential tail bounds on the likelihood ratio outside any fixed neighborhood of the true location parameter. These bounds are derived from the explicit form of the Pearson Type VII density and the fact that the tails are regularly varying with index greater than 1. The new lemma shows that, with probability tending to 1, no local maximum outside a shrinking neighborhood of the true value can exceed the value attained near the true parameter. Consequently, the global maximizer must coincide with the consistent root asymptotically. The revised manuscript incorporates this selection argument explicitly, thereby completing the strong-consistency claim for the MLE itself. revision: yes

Circularity Check

0 steps flagged

No circularity: standard MLE asymptotics derived from likelihood and general theorems

full rationale

The paper establishes strong consistency, asymptotic normality, Bahadur efficiency and asymptotic variance for the MLE of the location parameter in the Pearson Type VII family (including Cauchy) by applying standard limit theorems to the log-likelihood, while separately addressing the technical obstacle of multiple roots in the score equation. No quoted step defines a derived quantity in terms of itself, renames a fitted input as a prediction, or relies on a load-bearing self-citation whose content is unverified outside the present work. The derivation remains self-contained against external benchmarks such as Wald consistency arguments and uniform LLN, with the multiple-root handling constituting independent technical content rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard i.i.d. sampling model and the parametric form of the Pearson Type VII density with known scale; no additional free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)

domain assumption Observations are independent and identically distributed from a Pearson Type VII distribution with known scale and unknown location.
This defines the statistical model and the likelihood function under study.

pith-pipeline@v0.9.0 · 5587 in / 1280 out tokens · 29490 ms · 2026-05-18T01:26:19.782008+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.

We prove Theorem 1.1 by following the strategy of [4, Section 3.2] … Let Ln(t) := 1/n ∑ log(1+(Xi−t)2) … Cνm = {0} (Lemma 2.5).
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

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

The likelihood equation may have multiple roots; nevertheless, the maximum likelihood estimator performs well for large samples.

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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Forward citations

Cited by 1 Pith paper

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cs.IT 2026-05 unverdicted novelty 6.0

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

[1]

Bahadur efficiency of the max- imum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution.Ann

Yuichi Akaoka, Kazuki Okamura, and Yoshiki Otobe. Bahadur efficiency of the max- imum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution.Ann. Inst. Statist. Math., 74(5):895–923, 2022

work page 2022
[2]

Z. D. Bai and J. C. Fu. On the maximum-likelihood estimator for the location pa- rameter of a Cauchy distribution.Can. J. Stat., 15:137–146, 1987

work page 1987
[3]

Limit theorems for deviation means of independent and identically distributed random variables.J

Mátyás Barczy and Zsolt Páles. Limit theorems for deviation means of independent and identically distributed random variables.J. Theor. Probab., 36(3):1626–1666, 2023

work page 2023
[4]

Cambridge University Press, Cambridge, 2012

Abhishek Bhattacharya and Rabi Bhattacharya.Nonparametric inference on mani- folds With applications to shape spaces, volume 2 ofInstitute of Mathematical Statis- tics (IMS) Monographs. Cambridge University Press, Cambridge, 2012

work page 2012
[5]

Borwein and G

P. Borwein and G. Gabor. On the behavior of the MLE of the scale parameter of the Student family.Comm. Statist. A—Theory Methods, 13(24):3047–3057, 1984

work page 1984
[6]

Cohen Freue

Gabriela V. Cohen Freue. The Pitman estimator of the Cauchy location parameter. J. Statist. Plann. Inference, 137(6):1900–1913, 2007

work page 1900
[7]

R. A. Fisher. On the mathematical foundations of theoretical statistics.Philos. Trans. R. Soc. Lond., Ser. A, Contain. Pap. Math. Phys. Character, 222:309–368, 1922

work page 1922
[8]

Moderate deviations for the maximum likelihood estimator.Statist

Fuqing Gao. Moderate deviations for the maximum likelihood estimator.Statist. Probab. Lett., 55(4):345–352, 2001

work page 2001
[9]

Johnson, Samuel Kotz, and N

Norman L. Johnson, Samuel Kotz, and N. Balakrishnan.Continuous univariate dis- tributions. Vol. 2. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1995. A Wiley-Interscience Publication

work page 1995
[10]

Lange, Roderick J

Kenneth L. Lange, Roderick J. A. Little, and Jeremy M. G. Taylor. Robust statistical modeling using thetdistribution.J. Amer. Statist. Assoc., 84(408):881–896, 1989

work page 1989
[11]

Petrov.Limit theorems of probability theory, volume 4 ofOxford Stud- ies in Probability

Valentin V. Petrov.Limit theorems of probability theory, volume 4 ofOxford Stud- ies in Probability. The Clarendon Press, Oxford University Press, New York, 1995. Sequences of independent random variables, Oxford Science Publications

work page 1995
[12]

James A. Reeds. Asymptotic number of roots of Cauchy location likelihood equations. Ann. Statist., 13(2):775–784, 1985

work page 1985
[13]

Strong laws of large numbers for generalizations of Fréchet mean sets.Statistics, 56(1):34–52, 2022

Christof Schötz. Strong laws of large numbers for generalizations of Fréchet mean sets.Statistics, 56(1):34–52, 2022. 26 KAZUKI OKAMURA

work page 2022
[14]

Stephen J. Taylor. The variance of the maximum likelihood estimate of the shape parameter of the student distribution. Manuscript, Department of Operational Re- search, University of Lancaster, England, 1980

work page 1980
[15]

M. L. Tiku and R. P. Suresh. A new method of estimation for location and scale parameters.J. Stat. Plann. Inference, 30(2):281–292, 1992

work page 1992
[16]

David C. Vaughan. On the Tiku-Suresh method of estimation.Commun. Stat., The- ory Methods, 21(2):451–469, 1992

work page 1992
[17]

Empirical Bayesian estimation of location of the Cauchy distribution.J

Jin Zhang. Empirical Bayesian estimation of location of the Cauchy distribution.J. Stat. Comput. Simulation, 84(6):1381–1385, 2014. Department of Mathematics, F aculty of Science, Shizuoka University, 836, Ohya, Suruga-ku, Shizuoka, 422-8529, JAPAN. Email address:okamura.kazuki@shizuoka.ac.jp

work page 2014

[1] [1]

Bahadur efficiency of the max- imum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution.Ann

Yuichi Akaoka, Kazuki Okamura, and Yoshiki Otobe. Bahadur efficiency of the max- imum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution.Ann. Inst. Statist. Math., 74(5):895–923, 2022

work page 2022

[2] [2]

Z. D. Bai and J. C. Fu. On the maximum-likelihood estimator for the location pa- rameter of a Cauchy distribution.Can. J. Stat., 15:137–146, 1987

work page 1987

[3] [3]

Limit theorems for deviation means of independent and identically distributed random variables.J

Mátyás Barczy and Zsolt Páles. Limit theorems for deviation means of independent and identically distributed random variables.J. Theor. Probab., 36(3):1626–1666, 2023

work page 2023

[4] [4]

Cambridge University Press, Cambridge, 2012

Abhishek Bhattacharya and Rabi Bhattacharya.Nonparametric inference on mani- folds With applications to shape spaces, volume 2 ofInstitute of Mathematical Statis- tics (IMS) Monographs. Cambridge University Press, Cambridge, 2012

work page 2012

[5] [5]

Borwein and G

P. Borwein and G. Gabor. On the behavior of the MLE of the scale parameter of the Student family.Comm. Statist. A—Theory Methods, 13(24):3047–3057, 1984

work page 1984

[6] [6]

Cohen Freue

Gabriela V. Cohen Freue. The Pitman estimator of the Cauchy location parameter. J. Statist. Plann. Inference, 137(6):1900–1913, 2007

work page 1900

[7] [7]

R. A. Fisher. On the mathematical foundations of theoretical statistics.Philos. Trans. R. Soc. Lond., Ser. A, Contain. Pap. Math. Phys. Character, 222:309–368, 1922

work page 1922

[8] [8]

Moderate deviations for the maximum likelihood estimator.Statist

Fuqing Gao. Moderate deviations for the maximum likelihood estimator.Statist. Probab. Lett., 55(4):345–352, 2001

work page 2001

[9] [9]

Johnson, Samuel Kotz, and N

Norman L. Johnson, Samuel Kotz, and N. Balakrishnan.Continuous univariate dis- tributions. Vol. 2. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1995. A Wiley-Interscience Publication

work page 1995

[10] [10]

Lange, Roderick J

Kenneth L. Lange, Roderick J. A. Little, and Jeremy M. G. Taylor. Robust statistical modeling using thetdistribution.J. Amer. Statist. Assoc., 84(408):881–896, 1989

work page 1989

[11] [11]

Petrov.Limit theorems of probability theory, volume 4 ofOxford Stud- ies in Probability

Valentin V. Petrov.Limit theorems of probability theory, volume 4 ofOxford Stud- ies in Probability. The Clarendon Press, Oxford University Press, New York, 1995. Sequences of independent random variables, Oxford Science Publications

work page 1995

[12] [12]

James A. Reeds. Asymptotic number of roots of Cauchy location likelihood equations. Ann. Statist., 13(2):775–784, 1985

work page 1985

[13] [13]

Strong laws of large numbers for generalizations of Fréchet mean sets.Statistics, 56(1):34–52, 2022

Christof Schötz. Strong laws of large numbers for generalizations of Fréchet mean sets.Statistics, 56(1):34–52, 2022. 26 KAZUKI OKAMURA

work page 2022

[14] [14]

Stephen J. Taylor. The variance of the maximum likelihood estimate of the shape parameter of the student distribution. Manuscript, Department of Operational Re- search, University of Lancaster, England, 1980

work page 1980

[15] [15]

M. L. Tiku and R. P. Suresh. A new method of estimation for location and scale parameters.J. Stat. Plann. Inference, 30(2):281–292, 1992

work page 1992

[16] [16]

David C. Vaughan. On the Tiku-Suresh method of estimation.Commun. Stat., The- ory Methods, 21(2):451–469, 1992

work page 1992

[17] [17]

Empirical Bayesian estimation of location of the Cauchy distribution.J

Jin Zhang. Empirical Bayesian estimation of location of the Cauchy distribution.J. Stat. Comput. Simulation, 84(6):1381–1385, 2014. Department of Mathematics, F aculty of Science, Shizuoka University, 836, Ohya, Suruga-ku, Shizuoka, 422-8529, JAPAN. Email address:okamura.kazuki@shizuoka.ac.jp

work page 2014