A general method of density estimation for associated random variables

Abstract

Let {X_n;n≥1} be a sequence of stationary associated random variables having a common marginal density function f(x). Let Φ_n(x, y), n=1,2,..., be a sequence of Borel-measurable functions defined on R². Let f_n(x)=1/nΣⁿ_k=1Φ_n(x, X_k) be the empirical density function. Here we study a set of sufficient conditions under which the probability Pr(sup_{a+δ≤x≤b-δ}|f_n(x)-f(x)|>ε→0 at an exponential rate as n → ∞ where the rate possibly depends on ε, δ and f and [a, b] is a finite or an infinite interval.