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.