KERNEL-TYPE DENSITY AND FAILURE RATE ESTIMATION FOR ASSOCIATED SEQUENCES

ISHA BAGAI^{1} AND B. L. S. PRAKASA RAO^{2}

1Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, New Delhi-110016, India 2Indian Statistical Institute, 203 Barrackpore Trunk Road, Calcutta 700 035, India

(Received August 22, 1991; revised August 25, 1994)

Abstract. Let {X_{n}, n ≥ 1} be a strictly stationary sequence of associated random variables defined on a probability space (Ω, B, P) with probability density function f(x) and failure rate function r(x) for X_{1}. Let f_{n}(x) be a kerneltype 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 F̄_{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.

Key words and phrases: Density estimator, failure-rate estimator, kernel estimators, associated sequences.