Robust M-Estimation for Spliced Power-Law Models Under Cusp-Type Non-Regularity
Abstract
Statistical modeling of heavy-tailed data with bounded or finite body behaviour often relies on composite or spliced distributions characterized by structural thresholds. Because the probability density function features a sharp directional shift (a cusp) at the join point, the classical log-likelihood function exhibits a discontinuity in its first derivative, causing standard Cramér-Rao regularity conditions and maximum likelihood frameworks to break down. To overcome this non-regularity, we develop a robust, log-transformed asymmetric Huber M-estimation framework for the structural threshold, coupled with conditional shape maximum likelihood estimators. We derive a closed-form parameter-balancing condition that analytically guarantees Fisher consistency under structural tail asymmetry. Furthermore, we establish the joint asymptotic normality of the hybrid Maximum Likelihood and M-estimator system and derive its explicit sandwich covariance structure. A computationally efficient profiled Z-estimation algorithm is implemented to reduce the joint estimation to a one-dimensional root-finding problem. Finite-sample properties are validated through Monte Carlo simulations, and the practical utility of the framework is demonstrated on the benchmark Danish Fire Insurance claims dataset.
Related
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BibTeX
@misc{balakrishna2026robust,
title = {{Robust M-Estimation for Spliced Power-Law Models Under Cusp-Type Non-Regularity}},
author = {N Balakrishna},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30624},
eprint = {2610.30624},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30624},
}
RIS
TY - UNPB
TI - Robust M-Estimation for Spliced Power-Law Models Under Cusp-Type Non-Regularity
AU - N Balakrishna
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30624
UR - https://arxiv.org/abs/2610.30624
AB - Statistical modeling of heavy-tailed data with bounded or finite body behaviour often relies on composite or spliced distributions characterized by structural thresholds. Because the probability density function features a sharp directional shift (a cusp) at the join point, the classical log-likelihood function exhibits a discontinuity in its first derivative, causing standard Cramér-Rao regularity conditions and maximum likelihood frameworks to break down. To overcome this non-regularity, we develop a robust, log-transformed asymmetric Huber M-estimation framework for the structural threshold, coupled with conditional shape maximum likelihood estimators. We derive a closed-form parameter-balancing condition that analytically guarantees Fisher consistency under structural tail asymmetry. Furthermore, we establish the joint asymptotic normality of the hybrid Maximum Likelihood and M-estimator system and derive its explicit sandwich covariance structure. A computationally efficient profiled Z-estimation algorithm is implemented to reduce the joint estimation to a one-dimensional root-finding problem. Finite-sample properties are validated through Monte Carlo simulations, and the practical utility of the framework is demonstrated on the benchmark Danish Fire Insurance claims dataset.
ER -