Extension of Likelihood Ratio Analysis Method to Skorokhod M1 Topology (with Application to Poissonian Smooth Change-Point Model)
Abstract
We extend the Ibragimov-Khasminskii likelihood ratio analysis method, in the one-dimensional parameter case, to a framework based on the Skorokhod M_1 topology. The proposed framework applies to a broad class of statistical models, including those for which the normalized likelihood ratio processes are continuous while the limiting likelihood ratio process is discontinuous, a setting outside the scope of classical approaches based on the uniform or Skorokhod J_1 topologies. We derive sufficient conditions for the weak convergence of likelihood ratio processes in the space of c{à}dl{à}g functions on RR vanishing at ±∞, endowed with the Skorokhod M_1 topology, and show how this convergence yields the asymptotic behavior of the maximum likelihood estimator. We also introduce new techniques for controlling the M_1 modulus of continuity. Although these techniques are particularly well suited to models in which all jumps and steep continuous transitions are in the same direction, they are of independent interest and may prove useful beyond this setting. As an application, we study a smooth change-point model for inhomogeneous Poisson processes in the fast regime, where the transition interval shrinks faster than 1/n. We establish that in this case the maximum likelihood estimator has the same asymptotic behavior (consistency, rate of convergence, limiting distribution and convergence of moments) as in the corresponding ''pure'' change-point model.
Related
- Mirror Likelihood for Modeling Manifold Data with Application on Correlation Time Series
- An infinite-dimensional Skorokhod problem with a diagonalization approach
- Volterra-Skorokhod Reflection and Fluid Limits of Loss Queues
- Direct Parallel–Hierarchical Transformation Based on Evolutionary Development Models: Application to Laser Beam Spot Image Processing with a Prospective Extension to Railway Track Infrastructure Maintenance
- Decompositions of CSBPs via Poissonian Intertwining
BibTeX
@misc{amiri2026extension,
title = {{Extension of Likelihood Ratio Analysis Method to Skorokhod $M_1$ Topology (with Application to Poissonian Smooth Change-Point Model)}},
author = {Arij Amiri and Sergueï Dachian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15585},
eprint = {2609.15585},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15585},
}
RIS
TY - UNPB
TI - Extension of Likelihood Ratio Analysis Method to Skorokhod M_1 Topology (with Application to Poissonian Smooth Change-Point Model)
AU - Arij Amiri
AU - Sergue{ï} Dachian
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15585
UR - https://arxiv.org/abs/2609.15585
AB - We extend the Ibragimov-Khasminskii likelihood ratio analysis method, in the one-dimensional parameter case, to a framework based on the Skorokhod M_1 topology. The proposed framework applies to a broad class of statistical models, including those for which the normalized likelihood ratio processes are continuous while the limiting likelihood ratio process is discontinuous, a setting outside the scope of classical approaches based on the uniform or Skorokhod J_1 topologies. We derive sufficient conditions for the weak convergence of likelihood ratio processes in the space of c{à}dl{à}g functions on RR vanishing at ±∞, endowed with the Skorokhod M_1 topology, and show how this convergence yields the asymptotic behavior of the maximum likelihood estimator. We also introduce new techniques for controlling the M_1 modulus of continuity. Although these techniques are particularly well suited to models in which all jumps and steep continuous transitions are in the same direction, they are of independent interest and may prove useful beyond this setting. As an application, we study a smooth change-point model for inhomogeneous Poisson processes in the fast regime, where the transition interval shrinks faster than 1/n. We establish that in this case the maximum likelihood estimator has the same asymptotic behavior (consistency, rate of convergence, limiting distribution and convergence of moments) as in the corresponding ''pure'' change-point model.
ER -