arxiv: 2604.02722 · v1 · submitted 2026-04-03 · 🧮 math.ST · stat.TH

Parameter Estimation of Incomplete Gamma Subordinators

Meena Sanjay Babulal , Sunil Kumar Gauttam , Aditya Maheshwari This is my paper

Pith reviewed 2026-05-13 18:54 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords incomplete gamma subordinatorparameter estimationfractional momentsmethod of momentsmaximum likelihood estimatorasymptotic normalityLévy processstochastic subordinator

0 comments

The pith

Fractional moments allow consistent parameter estimation for incomplete gamma subordinators whose ordinary moments are infinite.

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

The paper shows how to estimate parameters of incomplete gamma, epsilon-incomplete gamma, and truncated incomplete gamma subordinators. It replaces the standard method of moments with a version that uses fractional moments for the first two families, because their positive moments diverge. It also constructs the maximum likelihood estimator for the alpha parameter directly from the observed jump distribution and proves that this estimator is asymptotically normal. The resulting procedures give explicit ways to recover the model parameters from sampled paths of these Lévy subordinators.

Core claim

We estimate the parameters of InG, InG-ε and TInG subordinators. We modify the method of moments to rely on fractional moments for InG and InG-ε, apply ordinary moments to TInG, and obtain the maximum likelihood estimator for the parameter alpha of the first two families by using the jump distribution of the process; we further establish the asymptotic normality of this MLE.

What carries the argument

Fractional-order moments inserted into the method-of-moments equations, together with the likelihood constructed from the jump-size distribution of the subordinator.

If this is right

Parameters of InG and InG-ε subordinators become recoverable from finite samples even though all positive integer moments diverge.
The MLE for alpha is asymptotically normal, so standard errors and confidence intervals are available for large data sets.
TInG parameters can be estimated by the usual moment-matching formulas without fractional adjustments.
The estimators supply explicit fitting procedures for any data set that records the jumps of these subordinators.

Where Pith is reading between the lines

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

The same fractional-moment device could be tested on other Lévy subordinators whose moment-generating functions are infinite in a neighborhood of zero.
In applications that time-change Brownian motion or other diffusions, these estimators would allow direct calibration from high-frequency jump data.
Finite-sample bias of the fractional-moment estimators could be corrected by bootstrap or jackknife procedures not derived in the paper.

Load-bearing premise

The chosen fractional moments exist and remain finite, and the jump distribution of the subordinator is fully known or directly observable.

What would settle it

A Monte Carlo experiment in which the fractional-moment estimators do not converge to the true parameter values as the number of observed jumps grows, or in which the MLE confidence intervals fail to achieve the nominal coverage predicted by asymptotic normality.

read the original abstract

In this paper, we estimate the parameters of InG, InG-$\epsilon$ and TInG subordinators which have been studied by Babulal \textit{et al} (see \cite{babulal}). We have modified the method of moments technique to use fractional moments of the InG and InG-$\epsilon$ subordinator due to their infinite moments. For the TInG subordinator's parameter estimation, we have used the method of moments. We also compute the maximum likelihood estimator(MLE) for the parameter $\alpha$ of the InG and InG-$\epsilon$ subordinators using jump distribution of the process. We also discussed the asymptotic normality of MLE.

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 adapts fractional-moment MOM and jump-based MLE to the authors' own prior incomplete-gamma subordinators, but the MLE justification stays thin on sampling details and regularity conditions.

read the letter

The main takeaway is that this work takes the incomplete-gamma subordinators defined in the authors' earlier paper and supplies estimation recipes: fractional moments for the method of moments on the InG and InG-ε cases, ordinary moments for TInG, and an MLE for alpha built from the jump distribution plus a claim of asymptotic normality. That is the whole package. Fractional moments are a sensible patch when integer moments diverge, and linking the MLE to jumps respects the Lévy structure, so the basic ideas are not misguided. If the full derivations are present and correct, the recipes could be useful for anyone already fitting these specific processes. The soft spots sit mostly in the MLE section. The paper does not spell out the observation scheme—high-frequency sampling, fixed horizon, or continuous monitoring—nor does it address how small-jump accumulation is handled or state the regularity conditions needed for the usual LAN or martingale CLT arguments to apply. Without those pieces the asymptotic normality claim cannot be checked against standard Lévy-process results, and the estimator itself may not be well-defined from typical discrete data. The work is also tightly circular: the objects being estimated were introduced by the same authors, so the novelty reduces to applying off-the-shelf tools rather than deriving new ones. This paper is for a narrow audience already using these subordinators in stochastic modeling, perhaps in finance or risk. A reader looking for broader advances in Lévy estimation or new process classes will find little. The thinking is straightforward and the methods are standard, but the supporting detail for the central MLE claim is light. I would send it to peer review so that referees familiar with subordinators can check the missing derivations and ask for simulations; it is not strong enough for an automatic accept but deserves that level of scrutiny rather than a desk rejection.

Referee Report

3 major / 3 minor

Summary. The manuscript develops parameter estimation for Incomplete Gamma (InG), InG-ε, and Truncated InG (TInG) subordinators. It modifies the method of moments to fractional moments for InG and InG-ε (due to infinite ordinary moments), applies standard MOM to TInG, constructs an MLE for the parameter α from the jump distribution of the process, and discusses asymptotic normality of this MLE.

Significance. If the MLE is well-defined and the asymptotic normality holds under verifiable conditions, the work supplies practical estimation procedures for these specific subordinators, addressing the issue of infinite moments via fractional orders. This could aid inference in applications modeled by gamma-type Lévy processes, provided the estimators are shown to be consistent and the observation scheme is made explicit.

major comments (3)

[MLE derivation] MLE section: the likelihood is stated to be built directly from the jump distribution, yet no explicit expression for the log-likelihood is given, no observation scheme (continuous monitoring, high-frequency sampling with truncation, or fixed-horizon discrete observations) is specified, and no handling of the infinite small-jump activity of the Lévy measure appears. Without these, the MLE is not demonstrably computable from typical data.
[Asymptotics of MLE] Asymptotic normality discussion: the claim that the MLE for α is asymptotically normal is made without stating the sampling regime (T→∞ or n→∞), without regularity conditions on the Lévy measure (e.g., integrability near zero or positive-definiteness of the Fisher information), and without a proof sketch or reference to LAN/martingale CLT results for subordinators. This renders the normality assertion unverifiable.
[Method of moments] Modified MOM section: the specific fractional orders chosen for the moments of InG and InG-ε are not justified by existence proofs or identifiability arguments; it is unclear whether these orders yield a consistent system of equations for all parameters or whether the resulting estimators remain well-behaved as the fractional order varies.

minor comments (3)

[Abstract] Abstract and introduction: the notation InG-ε is introduced without an immediate definition of ε or its admissible range; a brief parenthetical clarification would improve readability.
[References] References: the citation to Babulal et al. should include the full bibliographic details (arXiv number or journal) so readers can locate the prior definitions of the subordinators.
[Notation] Notation consistency: ensure that the parameter vector (α, β, …) is uniformly denoted across the MOM and MLE sections to avoid confusion when comparing estimators.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. These have highlighted areas where additional rigor and clarity are needed. We address each major comment point by point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [MLE derivation] MLE section: the likelihood is stated to be built directly from the jump distribution, yet no explicit expression for the log-likelihood is given, no observation scheme (continuous monitoring, high-frequency sampling with truncation, or fixed-horizon discrete observations) is specified, and no handling of the infinite small-jump activity of the LÃ©vy measure appears. Without these, the MLE is not demonstrably computable from typical data.

Authors: We acknowledge that the MLE section requires more explicit details. In the revised manuscript we will derive and state the explicit log-likelihood expression constructed from the jump distribution for the InG and InG-Îµ subordinators. We will specify the observation scheme as high-frequency sampling over a fixed time horizon T with a truncation threshold chosen to handle the infinite small-jump activity of the LÃ©vy measure, following standard approaches for infinite-activity LÃ©vy processes. These additions will make the MLE demonstrably computable from typical data. revision: yes
Referee: [Asymptotics of MLE] Asymptotic normality discussion: the claim that the MLE for Î± is asymptotically normal is made without stating the sampling regime (Tâ†’âˆž or nâ†’âˆž), without regularity conditions on the LÃ©vy measure (e.g., integrability near zero or positive-definiteness of the Fisher information), and without a proof sketch or reference to LAN/martingale CLT results for subordinators. This renders the normality assertion unverifiable.

Authors: We agree the asymptotic normality claim needs supporting details. In the revision we will explicitly state the sampling regime (high-frequency observations with nâ†’âˆž as T is fixed or Tâ†’âˆž under suitable discretization). We will list the required regularity conditions on the LÃ©vy measure, including integrability near zero and positive-definiteness of the Fisher information. A brief proof sketch will be added, invoking the martingale central limit theorem for subordinators together with a reference to LAN results for LÃ©vy processes under the stated conditions. revision: yes
Referee: [Method of moments] Modified MOM section: the specific fractional orders chosen for the moments of InG and InG-Îµ are not justified by existence proofs or identifiability arguments; it is unclear whether these orders yield a consistent system of equations for all parameters or whether the resulting estimators remain well-behaved as the fractional order varies.

Authors: The fractional orders were chosen to ensure finite moments given the infinite ordinary moments of InG and InG-Îµ. In the revised manuscript we will add a justification subsection proving existence of fractional moments of order Î² (0 < Î² < 1) from the tail properties of the underlying gamma distribution. We will also supply identifiability arguments showing that the resulting system of equations uniquely determines all parameters, and we will include a brief analysis of estimator behavior and consistency as the fractional order varies within the admissible range. revision: yes

Circularity Check

0 steps flagged

No significant circularity in estimation procedures

full rationale

The paper cites Babulal et al. solely for the definitions of the InG, InG-ε and TInG subordinators. The central contributions—modified method of moments via fractional moments, MLE construction for α from the jump distribution, and discussion of asymptotic normality—are developed as new applications of standard statistical techniques to those processes. No equation or claim reduces by construction to the prior definitions, no fitted input is relabeled as a prediction, and no uniqueness or ansatz is smuggled via self-citation. The derivation chain remains independent of the self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper rests on the existence and properties of the InG, InG-ε, and TInG subordinators as defined in the authors' prior work; no new free parameters, axioms, or invented entities are introduced in the estimation step itself.

pith-pipeline@v0.9.0 · 5416 in / 1163 out tokens · 44749 ms · 2026-05-13T18:54:31.744951+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.

Cost.FunctionalEquation washburn_uniqueness_aczel unclear

?

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

We have modified the method of moments technique to use fractional moments of the InG and InG-ε subordinator due to their infinite moments.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

S. M. Aljeddani and M. Mohammed. Parameter estimation of a model using maximum likelihood func- tion and bayesian analysis through moment of order statistics.Alexandria Engineering Journal, 75:221– 232, 2023

work page 2023
[2]

M. S. Babulal, S. K. Gauttam, and A. Maheshwari. L´ evy processes with jumps governed by lower incomplete gamma subordinator and its variations.Theory of Probability & Its Applications, 70(1):73– 91, 2025

work page 2025
[3]

Beghin and C

L. Beghin and C. Ricciuti. L´ evy processes linked to the lower-incomplete gamma function.Fractal and Fractional, 5(3), 2021

work page 2021
[4]

Bhuvaneswari, G

M. Bhuvaneswari, G. N. Balaji, and F. M. Al-turjman. Machine learning parameter estimation in a smart-city paradigm for the medical field.Smart Cities Performability, Cognition, & Security, 2019

work page 2019
[5]

Cahoy and F

D. Cahoy and F. Polito. Parameter estimation for fractional birth and fractional death processes.Sta- tistics and Computing, 24:1–12, 03 2012

work page 2012
[6]

D. O. Cahoy and F. Polito. Simulation and estimation for the fractional yule process.Methodology and Computing in Applied Probability, 14:383–403, 2012

work page 2012
[7]

D. O. Cahoy, V. V. Uchaikin, and W. A. Woyczynski. Parameter estimation for fractional Poisson processes.J. Statist. Plann. Inference, 140(11):3106–3120, 2010

work page 2010
[8]

Chalana, D

V. Chalana, D. R. Haynor, P. D. Sampson, and Y. Kim. Parameter estimation in deformable models using markov chain monte carlo. InMedical Imaging, 1997

work page 1997
[9]

S. Haug, C. Kl¨ uppelberg, A. Lindner, and M. Zapp. Method of moment estimation in the cogarch(1,1) model.The Econometrics Journal, 10(2):320–341, 06 2007

work page 2007
[10]

A. W. Lo. Maximum likelihood estimation of generalized itˆ o processes with discretely sampled data. Econometric Theory, 4(2):231–247, 1988

work page 1988
[11]

an estimation procedure for mixtures of distributions

P. D. M. Macdonald. Comments and queries comment on “an estimation procedure for mixtures of distributions” by choi and bulgren.Journal of the Royal Statistical Society: Series B (Methodological), 33(2):326–329, 12 2018

work page 2018
[12]

J. K. Murphy and S. J. Godsill. Bayesian parameter estimation of jump-langevin systems for trend following in finance.2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 4125–4129, 2015

work page 2015
[13]

Saleh and V

A. Saleh and V. Rohatgi.An Introduction to Probability and Statistics. 09 2015. Email address: a19pmt002@lnmiit.ac.in, bsgauttam@lnmiit.ac.in, cadityam@iimidr.ac.in ∗Department of Mathematics, The LNM Institute of Information Technology, Rupa ki Nangal, Post-Sumel, Via-Jamdoli Jaipur 302031, Rajasthan, India. 11 #Operations Management and Quantitative Tec...

work page 2015