Asymptotic Properties of a Statistical Estimator of the Jeffreys Divergence: The Case of Discrete Distributions
Vladimir Glinskiy,
Artem Logachov,
Olga Logachova,
Helder Rojas,
Lyudmila Serga,
Anatoly Yambartsev
Affiliations
Vladimir Glinskiy
Department of Business Analytics, Siberian Institute of Management—Branch of the Russian Presidential Academy of National Economy and Public Administration, Novosibirsk State University of Economics and Management, 630102 Novosibirsk, Russia
Artem Logachov
Department of Business Analytics, Siberian Institute of Management—Branch of the Russian Presidential Academy of National Economy and Public Administration, Novosibirsk State University of Economics and Management, 630102 Novosibirsk, Russia
Olga Logachova
Department of Higher Mathematics, Siberian State University of Geosystems and Technologies (SSUGT), 630108 Novosibirsk, Russia
Helder Rojas
Escuela Profesional de Ingeniería Estadística, Universidad Nacional de Ingeniería, Lima 00051, Peru
Lyudmila Serga
Department of Business Analytics, Siberian Institute of Management—Branch of the Russian Presidential Academy of National Economy and Public Administration, Novosibirsk State University of Economics and Management, 630102 Novosibirsk, Russia
Anatoly Yambartsev
Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo (USP), São Paulo 05508-220, Brazil
We investigate the asymptotic properties of the plug-in estimator for the Jeffreys divergence, the symmetric variant of the Kullback–Leibler (KL) divergence. This study focuses specifically on the divergence between discrete distributions. Traditionally, estimators rely on two independent samples corresponding to two distinct conditions. However, we propose a one-sample estimator where the condition results from a random event. We establish the estimator’s asymptotic unbiasedness (law of large numbers) and asymptotic normality (central limit theorem). Although the results are expected, the proofs require additional technical work due to the randomness of the conditions.
