A Second-Order Extension of Hájek's Convolution Theorem with Statistical Applications
Abstract
For a class of regular estimators, Hájek, in his celebrated ``Convolution Theorem,'' showed that the asymptotic distribution of a regular estimator is the convolution of the distribution of an efficient estimator and some residual distribution. This result constitutes the foundation of the concept of asymptotic efficiency of regular estimators. In this paper, we provide a second-order version of that classical result. Introducing a class of second-order regular estimators with a valid Edgeworth expansion, we derive their asymptotic distribution under contiguous alternatives and show that it is the convolution of the second-order efficient distribution and some second-order residual distribution. This constitutes a second-order extension of Hájek's convolution theorem. Based on this, we introduce a concept of {\it second-order robustness} for second-order regular estimators. For a class of general Bayes estimators and minimum contrast estimators in time series models, this second-order robustness is used (i) in the characterization of second-order robust priors, (ii) in a comparison between the second-order robustness of maximum likelihood and Whittle estimators.
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BibTeX
@misc{hirukawa2026secondorder,
title = {{A Second-Order Extension of Hájek's Convolution Theorem with Statistical Applications}},
author = {Junichi Hirukawa and Masanobu Taniguchi and Marc Hallin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32718},
eprint = {2609.32718},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32718},
}
RIS
TY - UNPB
TI - A Second-Order Extension of Hájek's Convolution Theorem with Statistical Applications
AU - Junichi Hirukawa
AU - Masanobu Taniguchi
AU - Marc Hallin
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32718
UR - https://arxiv.org/abs/2609.32718
AB - For a class of regular estimators, Hájek, in his celebrated ``Convolution Theorem,'' showed that the asymptotic distribution of a regular estimator is the convolution of the distribution of an efficient estimator and some residual distribution. This result constitutes the foundation of the concept of asymptotic efficiency of regular estimators. In this paper, we provide a second-order version of that classical result. Introducing a class of second-order regular estimators with a valid Edgeworth expansion, we derive their asymptotic distribution under contiguous alternatives and show that it is the convolution of the second-order efficient distribution and some second-order residual distribution. This constitutes a second-order extension of Hájek's convolution theorem. Based on this, we introduce a concept of {\it second-order robustness} for second-order regular estimators. For a class of general Bayes estimators and minimum contrast estimators in time series models, this second-order robustness is used (i) in the characterization of second-order robust priors, (ii) in a comparison between the second-order robustness of maximum likelihood and Whittle estimators.
ER -