###
KERNEL-TYPE DENSITY AND FAILURE RATE ESTIMATION

FOR ASSOCIATED SEQUENCES

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

^{1} *Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, New Delhi-110016, India*

^{2} *Indian 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 (*Omega*, \cal B,
\cal 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
\bar *F*_{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.

**Source**
( TeX ,
DVI ,
PS )