KERNEL ESTIMATES UNDER ASSOCIATION - STRONG UNIFORM CONSISTENCY

Let X1, X2,... be associated random variables forming a strictly stationary sequence, and let f be the probability density function of X1. For r greater-than-or-equal-to 0 integer, let f(r) be the r th order derivative of f. Under suitable regularity conditions on a kernel function K, a sequence of bandwidths {h(n)}, the derivatives f(s), s = 0, 1,..., r, and the covariances Cov( X1, X(i)), i greater-than-or-equal-to 2, the usual kernel estimate of f(r)(x) is shown to be strongly consistent, uniformly in x. An application is also presented in the estimation of the hazard rate. Finally, certain covergence rates are also discussed.