ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 1, pp. 22–37.
Pleiades Publishing, Inc., 2010.
Original Russian Text
A.V. Kitaeva, G.M. Koshkin, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 1, pp. 25–41.
METHODS OF SIGNAL PROCESSING
Nonparametric Semirecursive Identiﬁcation
in a Wide Sense of Strong Mixing Processes
A. V. Kitaeva
and G. M. Koshkin
Tomsk Polytechnical University
Tomsk State University
Department of Informatization Problems, Tomsk Scientiﬁc Center,
Siberian Branch of the RAS, Tomsk
Received January 21, 2009; in ﬁnal form, November 13, 2009
Abstract—We ﬁnd principal parts of asymptotic mean-square errors of semirecursive nonpara-
metric estimators of functionals of a multidimensional density function under the assumption
that observations satisfy a strong mixing condition. Results are illustrated by an example of a
nonlinear autoregression process.
be a sequence of random variables deﬁned on a probability space (Ω, F, P), and let
a σ-algebra F
,i≤ k ≤ j} be generated by (X
Deﬁnition 1. A strictly stationary sequence (X
satisﬁes the strong mixing condition (no-
|P(AB) − P(A) P(B)|↓0,τ→∞,τ>0. (1)
The parameter α(τ)isreferredtoasthestrong mixing coeﬃcient.
Time series satisfying condition (1) are often used in modeling economic, ﬁnancial, physical, and
technical processes [1–3].
The identiﬁcation problem in a wide sense for stochastic systems [4, 5] is often reduced to
estimating, based on observed sequences of output and input variables, functions of the form
(x)},i= 1,s, j = 1,m
where x ∈ R
, H(·): R
is a given function, a
a(x) ≡ a
(x)). Functionals a
(x) and their derivatives are deﬁned as follows:
(y)f (x, y) dy, a
,i= 1,s, j = 1,m, (3)
are known functions, f (x, y) is an unknown density function of the observed random
vector Z =(X, Y ) ∈ R
, X =(X
) are input variables, and Y is an output variable.
Integration in (3) is over the whole number axis, and in what follows we assume that
Supported in part by the Russian Foundation for Basic Research, project no. 09-08-00595a.