Nonparametric kernel estimation of Tsallis-type cumulative residual entropy under length-biased lifetime data

‎We develop nonparametric inference methods for Tsallis-based cumulative residual entropy functionals when dealing with length-biased lifetime observations. The paper introduces kernel-type estimators for both the static measure and its time-dependent version, with explicit corrections for the sampling mechanism that systematically oversamples longer-lived units. We derive large-sample approximations for bias and variance under standard smoothness assumptions and appropriate bandwidth choices, establish weak and L^۲-consistency, and prove central limit theorems. Numerical     experiments using exponential and Weibull distributions examine finite-sample behavior through bias, variance, and mean squared error calculations, while normality diagnostics validate the asymptotic approximations. We also apply the methodology to automotive component durability data, where results confirm the stable performance and practical     value in realistic length-biased scenarios where standard sampling assumptions break down.

کلیدواژه ها:

Asymptotic normality ، Tsallis entropy functionals ، Dynamic entropy measures ، Weighted kernel estimation ، Length-biased sampling