Strong uniform consistency of the mode estimator under α-mixing assumption and double truncation

This article investigates the strong uniform consistency of mode estimators under the α-mixing hypothesis and double truncation. It addresses the challenges posed by dependent samples, particularly in the context of weak dependence, and provides theoretical results that establish the consistency of the mode estimator. The variable of interest, is truncated by two variables, (left) and (right), where observations are available only when . The model assumes independence between and , with a non-truncation probability . The nonparametric mode estimation relies on a kernel-based density estimator , where the kernel is a probability density function, and the bandwidth ​ decreases as the sample size grows. Under regularity assumptions, theorems establish the uniform convergence of the density estimator and the mode estimator . The convergence rates are optimal and comparable to the complete and i.i.d. case. These results leverage specific conditions on the truncation structure and an optimal choice of the bandwidth ​. This study will be a valuable resource for scholars and practitioners interested in non-parametric kernel estimation methods for doubly truncated data.