Monotone representation and measurability of generalized $\psi$-estimators

Matyas Barczy; Zsolt P\'ales

arxiv: 2412.02783 · v2 · submitted 2024-12-03 · 🧮 math.ST · stat.TH

Monotone representation and measurability of generalized psi-estimators

Matyas Barczy , Zsolt P\'ales This is my paper

Pith reviewed 2026-05-23 08:12 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords generalized ψ-estimatormonotone representationmeasurabilityconvex optimizationmeasurable diagonalnonconvex optimizationstatistical estimator

0 comments

The pith

Generalized ψ-estimators can be constructed via a function ψ decreasing in its second variable, converting a nonconvex optimization into a convex one, and their measurability is equivalent to that of ψ in the first variable under a measured

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

The paper establishes a monotone representation for generalized ψ-estimators by using their unique existence to express them through a function ψ that decreases in its second argument. This representation converts the task of locating the estimator from a nonconvex optimization problem into an equivalent convex one. The authors further prove that, when the sample space possesses a measurable diagonal and ψ satisfies additional regularity conditions, the estimator itself is measurable precisely when ψ is measurable in its first variable. These results build directly on the existence theorem from the authors' earlier work and provide explicit conditions for both construction and measurability. A reader cares because the equivalence supplies a concrete route to build and verify the estimators in statistical contexts where measurability of the output is required.

Core claim

Applying the unique existence of a generalized ψ-estimator, the authors construct this estimator in terms of a function ψ which is decreasing in its second variable. They interpret the result as a bridge from a nonconvex optimization problem to a convex one. Supposing that the underlying measurable space has a measurable diagonal and some additional assumptions on ψ hold, the measurability of a generalized ψ-estimator is shown to be equivalent to the measurability of the corresponding function ψ in its first variable.

What carries the argument

The monotone representation of the generalized ψ-estimator constructed from a function ψ that decreases in its second variable.

If this is right

The problem of locating the estimator reduces to solving a convex optimization task equivalent to the original nonconvex formulation.
Measurability of the estimator can be verified directly by checking measurability of ψ in its first variable.
The representation applies in every setting where the unique existence of the generalized ψ-estimator has already been established.
The equivalence supplies a practical criterion for confirming that the estimator is a measurable map when the sample space satisfies the diagonal condition.

Where Pith is reading between the lines

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

The convex reformulation may permit direct application of standard convex solvers to compute the estimator in practice.
The measurability equivalence could simplify proofs that certain statistical procedures based on these estimators preserve measurability when composed with other measurable maps.
One could test the representation by deriving explicit closed-form expressions for the estimator when ψ belongs to common families used in robust estimation.

Load-bearing premise

The unique existence of the generalized ψ-estimator together with the sample space having a measurable diagonal and further regularity conditions on ψ.

What would settle it

An explicit pair consisting of a sample space with measurable diagonal and a function ψ for which the unique generalized ψ-estimator exists yet cannot be written in the claimed monotone decreasing form, or for which ψ is measurable in the first variable but the resulting estimator is not.

read the original abstract

We investigate the monotone representation and measurability of generalized $\psi$-estimators introduced by the authors in 2022. Our first main result, applying the unique existence of a generalized $\psi$-estimator, allows us to construct this estimator in terms of a function $\psi$, which is decreasing in its second variable. We then interpret this result as a bridge from a nonconvex optimization problem to a convex one. Further, supposing that the underlying measurable space (sample space) has a measurable diagonal and some additional assumptions on $\psi$, we show that the measurability of a generalized $\psi$-estimator is equivalent to the measurability of the corresponding function $\psi$ in its first variable.

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

The paper gives a monotone representation for generalized ψ-estimators that recasts the problem as convex optimization and an equivalence for measurability under the measurable-diagonal condition, both resting on the 2022 unique-existence result.

read the letter

The main points are the monotone representation of the generalized ψ-estimator via a ψ that decreases in its second argument, presented as a bridge from nonconvex to convex optimization, and the equivalence showing that the estimator is measurable exactly when ψ is measurable in its first variable, provided the sample space has a measurable diagonal and some regularity on ψ. Both results apply the unique-existence theorem from the authors' 2022 paper rather than re-proving it here. The work does a straightforward job of spelling out these two extensions and their interpretations. The representation may give a computational handle by moving to convex problems, and the measurability statement is a clean technical clarification for spaces with the diagonal property. The central limitation is the dependence on the 2022 theorem: the manuscript invokes it to obtain the new claims but supplies no reproduction of that proof or explicit verification that every hypothesis carries over without adjustment. The abstract contains no counterexamples, error bounds, or worked examples, so the tightness of the application cannot be checked from the given text. The measurable-diagonal assumption is standard in some settings but narrows the scope. This is targeted work for researchers already using or citing the 2022 definition of generalized ψ-estimators, especially those concerned with optimization reformulations or measurability questions in measure-theoretic statistics. A reader working in that narrow area would extract the two stated results and the optimization bridge. The paper is coherent on its own terms and formally grounded enough to deserve a serious referee who can inspect the invocation of the prior theorem. I would send it to peer review; the argument holds once the 2022 result is granted, and the new statements are precise enough to be worth checking in detail.

Referee Report

2 major / 1 minor

Summary. The manuscript investigates the monotone representation and measurability of generalized ψ-estimators introduced in the authors' 2022 paper. Applying the unique-existence result from that work, the first main claim constructs the estimator via a function ψ that is decreasing in its second argument and interprets the construction as a bridge from nonconvex to convex optimization. Under the additional hypothesis that the underlying measurable space has a measurable diagonal together with regularity conditions on ψ, the second main claim establishes equivalence between measurability of the estimator and measurability of ψ in its first variable.

Significance. If the claims hold, the monotone representation supplies a concrete reduction of a nonconvex problem to a convex one and the measurability equivalence supplies a practical criterion for verifying that the estimator is a measurable map. Both results build directly on the 2022 unique-existence theorem and therefore inherit whatever verification of hypotheses is supplied for the present class of ψ functions.

major comments (2)

[Statement of the first main result (Introduction / §2)] The first main result rests entirely on an application of the 2022 unique-existence theorem; the manuscript must therefore contain an explicit, item-by-item check that every hypothesis of that theorem is satisfied by the ψ functions under consideration here. Without this verification the monotone-representation claim is not self-contained.
[Measurability theorem (presumably §4)] The measurability equivalence is stated only under the measurable-diagonal hypothesis on the sample space; the paper should either prove that the hypothesis is necessary (by exhibiting a counter-example when it fails) or show that it can be replaced by a weaker, more easily checked condition.

minor comments (1)

The abstract refers to 'some additional assumptions on ψ' without listing them; these assumptions must be stated explicitly at the point where the measurability equivalence is announced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below.

read point-by-point responses

Referee: [Statement of the first main result (Introduction / §2)] The first main result rests entirely on an application of the 2022 unique-existence theorem; the manuscript must therefore contain an explicit, item-by-item check that every hypothesis of that theorem is satisfied by the ψ functions under consideration here. Without this verification the monotone-representation claim is not self-contained.

Authors: We agree that an explicit, item-by-item verification is required for self-containment. The revised manuscript will add a new subsection (immediately following the statement of the monotone-representation theorem) that enumerates each hypothesis of the 2022 unique-existence theorem and verifies it directly for the class of decreasing ψ functions used here. revision: yes
Referee: [Measurability theorem (presumably §4)] The measurability equivalence is stated only under the measurable-diagonal hypothesis on the sample space; the paper should either prove that the hypothesis is necessary (by exhibiting a counter-example when it fails) or show that it can be replaced by a weaker, more easily checked condition.

Authors: The measurable-diagonal assumption is invoked to obtain the equivalence. We will expand §4 with a short discussion of its role and will either (i) supply a counter-example on a space without measurable diagonal or (ii) identify a strictly weaker condition (e.g., the existence of a measurable selection on the diagonal) under which the equivalence still holds. If neither is feasible within the scope of the paper, we will state the limitation explicitly. revision: partial

Circularity Check

0 steps flagged

Minor self-citation of 2022 uniqueness theorem; new monotone and measurability results are independent derivations

full rationale

The manuscript invokes the unique existence theorem from the authors' own 2022 paper as the starting point for constructing a monotone representation (ψ decreasing in the second argument) and for proving the measurability equivalence under the measurable-diagonal hypothesis. This is a standard citation of a prior established result rather than any reduction of the new claims to fitted quantities or self-definitional loops within the present text. No equations or steps in the current work are shown to be equivalent to their inputs by construction, and the central claims retain independent mathematical content once the cited theorem is granted. Self-citation of this form does not trigger circularity under the evaluation rules.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger populated from abstract only; the central claims rest on the unique-existence statement from prior work and on the measurable-diagonal assumption.

axioms (2)

domain assumption Unique existence of a generalized ψ-estimator
Invoked explicitly as the basis for the first main result.
domain assumption Sample space has a measurable diagonal
Required for the measurability equivalence.

pith-pipeline@v0.9.0 · 5639 in / 1163 out tokens · 37038 ms · 2026-05-23T08:12:14.466189+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Barczy and Zs

M. Barczy and Zs. P´ ales. Existence and uniqueness of weighted generalized ψ -estimators. arXiv 2211.06026, 2022

work page arXiv 2022
[2]

Barczy and Zs

M. Barczy and Zs. P´ ales. Basic properties of generalized ψ -estimators. To appear in Publ. Math. Debrecen, 2025+. Available also at arXiv 2401.16127

work page arXiv 2025
[3]

V. I. Bogachev. Measure Theory. Vol. I, II . Springer-Verlag, Berlin, 2007

work page 2007
[4]

Dar´ oczy and Zs

Z. Dar´ oczy and Zs. P´ ales. On comparison of mean values. Publ. Math. Debrecen , 29(1-2):107– 115, 1982

work page 1982
[5]

Hewitt and K

E. Hewitt and K. Stromberg. Real and Abstract Analysis , volume No. 25 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, third edition, 1975

work page 1975
[6]

P. J. Huber. Robust estimation of a location parameter. Ann. Math. Statist. , 35:73–101, 1964

work page 1964
[7]

P. J. Huber. The behavior of maximum likelihood estimates under no nstandard conditions. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probabilit y (Berkeley, Calif., 1965/66), Vol. I: Statistics , pages 221–233. Univ. California Press, Berkeley, Calif., 1967

work page 1965
[8]

A. Klenke. Probability Theory – A Comprehensive Course . Universitext. Springer, Cham, third edition, 2020

work page 2020
[9]

M. R. Kosorok. Introduction to Empirical Processes and Semiparametric In ference. Springer Series in Statistics. Springer, New York, 2008

work page 2008
[10]

Zs. P´ ales. General inequalities for quasideviation means. Aequationes Math. , 36(1):32–56, 1988

work page 1988
[11]

A. W. van der Vaart. Asymptotic Statistics , volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 1998. 17

work page 1998

[1] [1]

Barczy and Zs

M. Barczy and Zs. P´ ales. Existence and uniqueness of weighted generalized ψ -estimators. arXiv 2211.06026, 2022

work page arXiv 2022

[2] [2]

Barczy and Zs

M. Barczy and Zs. P´ ales. Basic properties of generalized ψ -estimators. To appear in Publ. Math. Debrecen, 2025+. Available also at arXiv 2401.16127

work page arXiv 2025

[3] [3]

V. I. Bogachev. Measure Theory. Vol. I, II . Springer-Verlag, Berlin, 2007

work page 2007

[4] [4]

Dar´ oczy and Zs

Z. Dar´ oczy and Zs. P´ ales. On comparison of mean values. Publ. Math. Debrecen , 29(1-2):107– 115, 1982

work page 1982

[5] [5]

Hewitt and K

E. Hewitt and K. Stromberg. Real and Abstract Analysis , volume No. 25 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, third edition, 1975

work page 1975

[6] [6]

P. J. Huber. Robust estimation of a location parameter. Ann. Math. Statist. , 35:73–101, 1964

work page 1964

[7] [7]

P. J. Huber. The behavior of maximum likelihood estimates under no nstandard conditions. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probabilit y (Berkeley, Calif., 1965/66), Vol. I: Statistics , pages 221–233. Univ. California Press, Berkeley, Calif., 1967

work page 1965

[8] [8]

A. Klenke. Probability Theory – A Comprehensive Course . Universitext. Springer, Cham, third edition, 2020

work page 2020

[9] [9]

M. R. Kosorok. Introduction to Empirical Processes and Semiparametric In ference. Springer Series in Statistics. Springer, New York, 2008

work page 2008

[10] [10]

Zs. P´ ales. General inequalities for quasideviation means. Aequationes Math. , 36(1):32–56, 1988

work page 1988

[11] [11]

A. W. van der Vaart. Asymptotic Statistics , volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 1998. 17

work page 1998