Problems of Control and Information Theory 8. (Budapest, 1979)

1979 / 1. szám - Greblicki, W.: Nonparametric system identification by orthogonal series

GREBLICKI: NONPARAMETRIC SYSTEM IDENTIFICATION BY ORTHOGONAL SERIES f(x) ;> « > 0 Theorem 1. If Ф0(х) £ Z/2> EY2 < oo, (4) (5) (a is independent of x), all the functions of the orthonormal system are jointly bounded, i.e. \<Pi(%)\ <, c (6) (c is independent of i), and lim N(n) -- oo , lim N(n)/n = 0, (7) n-*~«» П~* °° then lim E J" (Ф0{х) — 0n(x))2dx =0. (8) Proof. It is clear that the estimator (3) is unbiased and its variance is bounded by Е{щ - ainf <, n~1E{ Y<pi(X)/f(X)}2 ^ n~^x~2c2EY2 = dn~\ By Fubini’s theorem and Parseval’s formula we have Convergence (8) is a consequence of the above inequality and (7). Thus, the theorem is proved. As it has been shown, in order that the optimal model may be expanded, its square integrability must be assumed. This condition can be neglected while expanding Ф0(x)fll2{x). It is square integrable, because of EY2 < oo. Thus, T0{x)fl2{x) ~ JZbjcpiix), 1=0 where bt = jФ0(х) Cfiix) f\x) dx = E{Y(pi{X) rll2(X)). Each bi will be estimated by bin = »-1i[Wr1/,W N(n)E [ (Ф0(х) — Фnix)')2 dx = ~ ^ dt <L 1=0 !=Л/(л) + 1 a?. i=W(n) + l

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