On the convergence rates of kernel estimator and hazard estimator for widely dependent samples

Abstract

In this paper, we establish a Bernstein-type inequality for widely orthant dependent random variables, and obtain the rates of strong convergence for kernel estimators of density and hazard functions, under some suitable conditions.

Complete f-moment convergence for weighted sums of WOD arrays with statistical applications

“…The concept of WOD random variables was introduced by Wang et al [27]. Since then, many scholars studied the probability limit properties of WOD random variables and obtained some interesting results; Wang et al [37] presented some probability inequalities and moment inequalities for WOD random variables, further got the complete convergence for weighted sums of arrays of rowwise WOD random variables and gave some applications; Qiu and Hu [15] investigated the strong limit theorems for weighted sums of WOD random variables; Qiu and Chen [13] established a complete convergence result and a complete moment convergence result for weighted sums of WOD random variables under mild conditions; Shen et al [20] provided some exponential probability inequalities to get the complete convergence for arrays of rowwise WNOD random variables; Wu et al [32] investigated complete moment convergence for WOD random variables under some mild conditions; Li et al [11] established a Bernstein-type inequality for WOD random variables, and obtain the rates of strong convergence for kernel estimators of density and hazard functions under some suitable conditions; He [17]established the strong consistency and complete consistency of the Priestley-Chao estimator in nonparametric regression model with WOD errors under some general conditions and obtained the rates of strong consistency and complete consistency; Lu et al [14] studied the complete f -moment convergence for WOD random variables and gave some applications; Chen and Sung [3] obtained a Spitzer-type law of large numbers for WOD random variables. Shen and Wu [19] investigated the complete qth moment convergence and provided some sufficient conditions for sums of WOD random variables; Xi et al [28] presented some convergence properties for partial sums of WOD random variables and gave some applications; Lang et al [18] investigated the complete convergence for weighted sums of WOD random variables, and so on.…”