Order of convergence estimate for a stochastic iterative method for the mode and its size

Abstract

Finding the mode of an unknown distribution using only the samples is a classical statistical problem. For skewed data, the metric mode offers better insight than the other central tendencies, such as the mean and median. Developing a sample-based recursive method to estimate the mode and studying its convergence analysis is an interesting problem in stochastic approximation. This manuscript proposes a generalized recursive sample-based algorithm for mode estimation and a companion algorithm for finding the mode size. Furthermore, under suitable assumptions, the almost sure convergence of the proposed coupled iterative method, along with a sharper order of convergence, is derived using the theory developed in stochastic approximation. The performance of the proposed algorithm is compared with the existing iterative techniques in the literature, and its superior performance is demonstrated empirically. © 2026 Taylor & Francis Group, LLC.