AISM 52, 599-611

## Nonparametric density estimation for a long-range dependent linear process

### Toshio Honda

Institute of Social Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, JAPAN

(Received February 17, 1999; revised August 12, 1999)

Abstract.
We estimate the marginal density function of
a long-range dependent linear process by the kernel estimator.
We assume the innovations are i.i.d. Then it is known that
the term of the sample mean is dominant in the MISE of the
kernel density estimator when the dependence is beyond some level
which depends on the bandwidth and that the MISE has
asymptotically the same form as for i.i.d. observations when
the dependence is below the level.
We call the latter the case where the dependence is not very strong
and focus on it in this paper. We show that the asymptotic distribution
of the kernel density estimator is the same as for i.i.d. observations
and the effect of long-range dependence does not appear. In addition
we describe some results for weakly dependent linear processes.

Key words and phrases:
Kernel density estimator, long-range dependence, linear process, bandwidth, asymptotic normality.

**Source**
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