Density Estimation for a Class of Stationary Nonlinear Processes

Author： Chanda Kamal C.  

Publisher： Springer Publishing Company

ISSN： 0020-3157

Source： Annals of the Institute of Statistical Mathematics, Vol.55, Iss.1, 2003-03, pp. : 69-82

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Abstract

Let {Xt; t ∈ $${\Bbb Z}$$} be a strictly stationary nonlinear process of the form Xt = εt + Σr=1∞ Wrt, where Wrt can be written as a function gr(εt−1,…, εt−r−q), {εt; t ∈ $${\Bbb Z}$$} is a sequence of independent and identically distributed (i.i.d.) random variables with E|ε1|g < ∞ for some γ > 0 and q  0 is a fixed integer. Under certain mild regularity conditions on gr and {εt} we then show that X1 has a density function f and that the standard kernel type estimator $$\mathop f\limits^ \wedge _n (x)$$ based on a realization {X1,…, Xn} from {Xt} is, asymptotically, normal and converges a.s. to f(X) as n → ∞.

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