Density Estimates and Markov Sequences

Abstract

Estimates of the density function of a population based on a sample of independent observations have been considered in a number of papers [1,6-7]. Questions of bias, variance and asymptotic distribution of the estimates have been dealt with at greatest length. Our object is to look at such estimates of the density function when the observations are dependent. The results will not be dealt with in the most general context or under very general conditions. To obtain results in a simple and readily understandable form, the observations are assumed to be sampled from a stationary Markov sequence with a fairly strong condition on the Markov transition operator. However, the extent to which some of the conditions can be obviously relaxed will be indicated.