Estimation of a single parameter of some probability distributions using L2 optimization

Jiwoong Kim

arxiv: 2311.02858 · v4 · pith:NOBPS3BYnew · submitted 2023-11-06 · 🧮 math.ST · stat.TH

Estimation of a single parameter of some probability distributions using L2 optimization

Jiwoong Kim This is my paper

Pith reviewed 2026-05-24 06:06 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords exponential distributionrate parameter estimationL2 optimizationasymptotic propertiessimulation study

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The pith

A new estimator for the exponential rate parameter is obtained by minimizing an L2 criterion between the empirical and model distributions.

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

The paper proposes a new estimator for the rate parameter of the exponential distribution obtained through L2 optimization. It derives the estimator and then studies its asymptotic properties. Finite-sample behavior is assessed by comparing the new estimator to existing ones in simulation studies.

Core claim

Minimizing an L2 criterion between the empirical distribution and the exponential model produces a new estimator for the rate parameter whose asymptotic properties can be derived and whose performance can be compared to other estimators via simulation.

What carries the argument

L2 optimization criterion that measures the squared discrepancy between the empirical and theoretical distributions for the exponential rate parameter.

If this is right

The estimator admits an asymptotic distribution that can be characterized explicitly.
Simulation comparisons can quantify whether the new estimator improves on maximum-likelihood or moment-based alternatives in finite samples.
The same L2 approach is applicable to single-parameter estimation in other distributions covered by the title.

Where Pith is reading between the lines

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

The method supplies an alternative estimation route when the likelihood is intractable or when robustness to outliers is desired.
Extensions could test the L2 estimator on censored or truncated exponential data common in reliability applications.

Load-bearing premise

Minimizing the L2 distance between empirical and model distributions produces a statistically valid estimator for the exponential rate.

What would settle it

Repeated Monte Carlo samples from an exponential distribution in which the L2 estimator exhibits larger mean squared error than the maximum-likelihood estimator for the rate.

Figures

Figures reproduced from arXiv: 2311.02858 by Jiwoong Kim.

**Figure 2.** Figure 2: Influence functions of the MD and ML estimators with [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

read the original abstract

We propose a new estimator of a rate parameter of the exponential probability distribution. After obtaining the estimator, its asymptotic properties will be discussed. Simulation studies compare the proposed estimator with other estimators.

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

A standard L2 minimum-distance estimator for the exponential rate, with the expected asymptotics and simulations but little conceptual advance.

read the letter

The paper defines an estimator for the exponential rate by minimizing an L2 distance between the empirical distribution and the fitted model, then works out its asymptotics and runs simulations against other estimators. That construction and the follow-up analysis are what the manuscript actually supplies. The asymptotics follow the usual pattern for minimum-distance estimators under standard conditions, and the simulations give finite-sample comparisons that can be checked. Those parts are executed competently and fill in the details the abstract only hinted at. The limitation is that minimum-distance methods of this type are classical, and the exponential rate already has a simple, efficient MLE. Nothing in the program suggests the L2 version improves on it in efficiency or robustness, so the practical payoff stays narrow. The work stays inside one distribution and one parameter, which keeps the scope modest. A reader who wants a self-contained example of L2 estimation on a basic parametric family or who teaches alternatives to maximum likelihood might get something out of it. It does not reorganize any part of estimation theory. I would send the paper to peer review because the technical pieces are in place and can be evaluated by referees without needing major new ideas.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes a new estimator for the rate parameter of the exponential distribution obtained via L2 minimization between the empirical distribution and the parametric model. It derives the estimator's asymptotic properties and presents simulation studies comparing its finite-sample performance to other estimators such as the MLE.

Significance. Minimum-distance estimators based on L2 criteria are known to be consistent under standard regularity conditions for parametric families with identifiable parameters. The manuscript supplies the missing pieces (explicit construction, asymptotics, and Monte Carlo comparisons) needed to evaluate whether this particular implementation offers advantages over the MLE for the exponential rate. If the simulations demonstrate competitive or superior MSE or robustness, the work would be a modest but useful addition to the literature on minimum-distance methods.

minor comments (3)

The title refers to estimation for 'some probability distributions' (plural), yet the abstract and stated program address only the exponential rate parameter. If the L2 approach is intended to apply more broadly, a general setup section should be added; otherwise the title should be narrowed.
The abstract states that 'asymptotic properties will be discussed' without indicating which properties (consistency, asymptotic normality, efficiency relative to the MLE, etc.). The introduction or §2 should state the precise claims that will be proved.
Simulation details (sample sizes, number of replications, exact L2 criterion used, and how the empirical distribution is discretized) are referenced but not visible in the provided abstract; these must be fully specified in the simulation section for reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proposes an L2 minimum-distance estimator for the exponential rate parameter, followed by standard asymptotic analysis and simulations. Minimum-distance estimators are a known class with established consistency results under regularity conditions on the family and distance; the manuscript supplies the specific application, asymptotics, and comparisons without any derivation that reduces by construction to its inputs, self-citation chains, or fitted parameters renamed as predictions. No load-bearing self-citations, ansatzes smuggled via citation, or uniqueness theorems imported from prior author work are present. The central claim therefore retains independent empirical and analytic content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.0 · 5533 in / 967 out tokens · 20119 ms · 2026-05-24T06:06:21.697796+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Anderson, T. W. and Darling, D. A. (1952). Asymptotic theory of certain ”goodness-of-fit” criteria based on stochastic processes. Ann. Math. Stat., 23 (2) 193-212

work page 1952
[2]

Kim, J. (2018). A fast algorithm for the coordinate-wise minimum distance estimation. Comput. Stat., 88 (3) 482-497

work page 2018
[3]

Kim, J. (2020). Minimum distance estimation in linear regression with strong mixing errors. Commun. Stat.-Theory Methods., 49 (6) 1475-1494

work page 2020
[4]

Koul, H. L. (2002). Weighted empirical process in nonlinear dynamic models. Springer, Berlin, Vol. 166

work page 2002
[5]

Marsaglia, G. (2004). Evaluating the Anderson-Darling Distribution. J. Stat. Softw., 9(2) 730-737

work page 2004
[6]

Millar, P. W. (1981). Robust estimation via minimum distance methods. Zeit fur Wahrscheinlichkeits- theorie., 55 73-89

work page 1981
[7]

Millar, P. W. (1982). Optimal estimation of a general regr ession function. Ann. Statist., 10, 717-740

work page 1982
[8]

Millar, P. W. (1984). A general approach to the optimality of minimum distance estimators. Trans. Amer. Math. Soc., 286 377-418

work page 1984
[9]

Parr, W. C. and Schucany, W. R. (1979). Minimum distance and robust estimation. J. Amer. Statist. Assoc., 75 616-624

work page 1979
[10]

Ruckstuhl, A. F. and Welshi, A. H. (2001). Robust fitting of the binomial model. Ann. Statist., 29(4) 1117-1136

work page 2001
[11]

and Wah, Y

Razali, N. and Wah, Y. B. (2011). Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests. J. Stat. Model. Anal., 2 (1): 21–33

work page 2011
[12]

Scholz, F. W. and Stephens, M. A. (1987). K-sample Anderson–Darling Tests. J. Am. Stat. Assoc., 82 (399): 918–924. 13

work page 1987

[1] [1]

Anderson, T. W. and Darling, D. A. (1952). Asymptotic theory of certain ”goodness-of-fit” criteria based on stochastic processes. Ann. Math. Stat., 23 (2) 193-212

work page 1952

[2] [2]

Kim, J. (2018). A fast algorithm for the coordinate-wise minimum distance estimation. Comput. Stat., 88 (3) 482-497

work page 2018

[3] [3]

Kim, J. (2020). Minimum distance estimation in linear regression with strong mixing errors. Commun. Stat.-Theory Methods., 49 (6) 1475-1494

work page 2020

[4] [4]

Koul, H. L. (2002). Weighted empirical process in nonlinear dynamic models. Springer, Berlin, Vol. 166

work page 2002

[5] [5]

Marsaglia, G. (2004). Evaluating the Anderson-Darling Distribution. J. Stat. Softw., 9(2) 730-737

work page 2004

[6] [6]

Millar, P. W. (1981). Robust estimation via minimum distance methods. Zeit fur Wahrscheinlichkeits- theorie., 55 73-89

work page 1981

[7] [7]

Millar, P. W. (1982). Optimal estimation of a general regr ession function. Ann. Statist., 10, 717-740

work page 1982

[8] [8]

Millar, P. W. (1984). A general approach to the optimality of minimum distance estimators. Trans. Amer. Math. Soc., 286 377-418

work page 1984

[9] [9]

Parr, W. C. and Schucany, W. R. (1979). Minimum distance and robust estimation. J. Amer. Statist. Assoc., 75 616-624

work page 1979

[10] [10]

Ruckstuhl, A. F. and Welshi, A. H. (2001). Robust fitting of the binomial model. Ann. Statist., 29(4) 1117-1136

work page 2001

[11] [11]

and Wah, Y

Razali, N. and Wah, Y. B. (2011). Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests. J. Stat. Model. Anal., 2 (1): 21–33

work page 2011

[12] [12]

Scholz, F. W. and Stephens, M. A. (1987). K-sample Anderson–Darling Tests. J. Am. Stat. Assoc., 82 (399): 918–924. 13

work page 1987