KERNEL-TYPE DENSITY AND FAILURE RATE ESTIMATION FOR ASSOCIATED SEQUENCES

Let {X(n), n greater than or equal to 1} be a strictly stationary sequence of associated random variables defined on a probability space (Omega, B, P) with probability density function f(x) and failure rate function r(x) for X(1). Let f(n)(x) be a kernel-type estimator of f(x) based on X(1),...,X(n). Properties of f(n)(x) are studied. Pointwise strong consistency and strong uniform consistency are established under a certain set of conditions. An estimator r(n)(x) of r(x) based on f(n)(x) and (F) over bar(n)(x), the empirical survival function, is proposed. The estimator r(n)(x) is shown to be pointwise strongly consistent as well as uniformly strongly consistent over some sets.