Density estimation for a class of stationary nonlinear processes

Abstract

Let {X t ;t∈ℤ be a strictly stationary nonlinear process of the formX t =e t +∑ r=1 ∞ W rt , whereW rt can be written as a functiong r (e t−1,...e t-r-q ), {e t ;t∈ℤ is a sequence of independent and identically distributed (i.i.d.) random variables withE|e1| g 0 andq≥0 is fixed integer. Under certain mild regularity conditions ofg r and {e t } we then show thatX 1 has a density functionf and that the standard kernel type estimator\(\hat f_n (x)\) baded on a realization {X 1,...,X n } from {X t } is, asymptotically, normal and converges a.s. tof(x) asn→∞.