Robust estimation problems for stochastic processes

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Hauptverfasser:	Moklyachuk, M., Masyutka, A.
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institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Robust estimation problems for stochastic processes
spellingShingle	Robust estimation problems for stochastic processes Moklyachuk, M. Masyutka, A.
title_short	Robust estimation problems for stochastic processes
title_full	Robust estimation problems for stochastic processes
title_fullStr	Robust estimation problems for stochastic processes
title_full_unstemmed	Robust estimation problems for stochastic processes
title_sort	robust estimation problems for stochastic processes
author	Moklyachuk, M. Masyutka, A.
author_facet	Moklyachuk, M. Masyutka, A.
publishDate	2006
language	English
publisher	Інститут математики НАН України
format	Article
issn	0321-3900
url	https://nasplib.isofts.kiev.ua/handle/123456789/4460
citation_txt	Robust estimation problems for stochastic processes / M. Moklyachuk, A. Masyutka // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 88–113. — Бібліогр.: 22 назв.— англ.
work_keys_str_mv	AT moklyachukm robustestimationproblemsforstochasticprocesses AT masyutkaa robustestimationproblemsforstochasticprocesses
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last_indexed	2025-11-26T01:40:53Z
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fulltext	Theory of Stochastic Processes Vol. 12 (28), no. 3–4, 2006, pp. 88–113 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA ROBUST ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES1 We deal with the problem of optimal linear estimation of the functional AL �ξ = ∫ L 0 �a(t)�ξ(t)dt which depends on the unknown values of a multi- dimensional stationary stochastic process �ξ(t) with the spectral density F (λ) based on observations of the process �ξ(t) + �η(t) for t ∈ R\[0, L], where �η(t) is uncorrelated with �ξ(t) multidimensional stationary process with the spectral density G(λ) (interpolation problem), and the prob- lem of optimal linear estimation of the functional A�ξ = ∫∞ 0 �a(t)�ξ(t)dt which depends on the unknown values of a multidimensional stationary stochastic process �ξ(t), t ≥ 0, from observations of the process �ξ(t)+�η(t) for t < 0 (extrapolation problem). Formulas are obtained for calculation the mean square errors and the spectral characteristics of the optimal estimates of the functionals under the condition that the spectral density matrix F (λ) of the signal process �ξ(t) and the spectral density matrix G(λ) of the noise process �η(t) are known. The least favorable spectral densities and the minimax spectral characteristics of the optimal esti- mates of the functionals are found for concrete classes D = DF × DG of spectral densities under the condition that spectral density matrices F (λ) and G(λ) are not known, but classes D = DF × DG of admissible spectral densities are given. 1. Introduction Traditional methods of solution of the linear extrapolation, interpolation and filtering problems for stationary stochastic processes may be employed under the condition that spectral densities of processes are known exactly (see, for example, selected works of A. N. Kolmogorov (1992), survey arti- cle by T. Kailath (1974), books Yu. A. Rozanov (1990), N. Wiener (1966) and A. M. Yaglom (1987)). In practice, however, complete information on the spectral densities is impossible in most cases. To solve the problem one finds parametric or nonparametric estimates of the unknown spectral densities or selects these densities by other reasoning. Then applies the classical estimation method provided that the estimated or selected density is the true one. This procedure can result in a significant increasing of the 1Invited lecture. 2000 Mathematics Subject Classification. Primary 60G10, 62M20, 60G35, 93E10, 93E11. Key words and phrases. Stationary stochastic process, interpolation, extrapolation, filtering, robust estimate, observations with noise, mean square error, least favorable spectral densities, minimax spectral characteristic. 88 ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 89 value of error as K. S. Vastola and H. V. Poor (1983) have demonstrated with the help of some examples. This is a reason to search estimates which are optimal for all densities from a certain class of the admissible spectral densities. These estimates are called minimax since they minimize the max- imal value of the error. A survey of results in minimax (robust) methods of data processing can be found in the paper by S. A. Kassam and H. V. Poor (1985). The paper by Ulf Grenander (1957) should be marked as the first one where the minimax approach to extrapolation problem for stationary processes was proposed. J. Franke (1984, 1985, 1991), J. Franke and H. V. Poor (1984) investigated the minimax extrapolation and filtering prob- lems for stationary sequences with the help of convex optimization methods. This approach makes it possible to find equations that determine the least favorable spectral densities for various classes of densities. In the papers by M. P. Moklyachuk (1994, 1997, 1998, 2000, 2001), M. P. Moklyachuk and A. Yu. Masyutka (2005, 2006) the minimax approach to extrapolation, interpolation and filtering problems are investigated for functionals which depend on the unknown values of stationary processes and sequences. In this article we considered the problem of estimation of the unknown value of the functional AL �ξ = ∫ L 0 �a(t)�ξ(t)dt which depends on the unknown values of a multidimensional stationary stochastic process �ξ(t) = {ξk(t)}T k=1 , E�ξ(t) = 0, with the spectral density matrix F (λ) = {fij(λ)}T i,j=1 based on observations of the process �ξ(t) + �η(t) for t ∈ R\[0, L], where �η(t) = {ηk(t)}T k=1 is an uncorrelated with �ξ(t) multidimensional stationary sto- chastic process with the spectral density matrix G(λ) = {gij(λ)}T i,j=1 (in- terpolation problem), and the problem of optimal linear estimation of the functional A�ξ = ∫∞ 0 �a(t)�ξ(t)dt which depends on the unknown values of a multidimensional stationary stochastic process �ξ(t), t ≥ 0, from observa- tions of the process �ξ(t) + �η(t) for t < 0 (extrapolation problem). Formulas are obtained for calculation the mean square errors and the spectral char- acteristics of the optimal linear estimates of the functionals under condition that the spectral density matrices F (λ) and G(λ) are known. Formulas are proposed that determine the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of the func- tionals for concrete classes D = DF × DG of spectral densities under the condition that spectral density matrices F (λ), G(λ) are not known, but classes D = DF × DG of admissible spectral densities are given. 2. Hilbert space projection method of interpolation Let �ξ(t) = {ξk(t)}T k=1 , E�ξ(t) = 0, �η(t) = {ηk(t)}T k=1, E�η(t) = 0, be uncor- related multidimensional stationary stochastic processes with the spectral 90 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA density matrices F (λ) = {fkl(λ)}T k,l=1 and G(λ) = {gkl(λ)}T k,l=1 , which sat- isfy the minimality condition (1) ∞∫ −∞ b(λ)(F (λ) + G(λ))−1b∗(λ)dλ < ∞ for a nontrivial vector function of the exponential type b(λ) = ∫ L 0 �α(t)eitλdt, where �α(t) = {αk(t)}T k=1 . Under this condition the error-free interpolation is impossible (see, for example, Yu. A. Rozanov (1990)). Denote by L2 (F ) the Hilbert space of vector-valued functions ϕ(λ) = {ϕk(λ)}T k=1 which are square integrable with respect to measure with the density F (λ) = {fkl(λ)}T k,l=1: ∞∫ −∞ ϕ(λ)F (λ)ϕ∗(λ)dλ = ∞∫ −∞ T∑ k,l=1 ϕk(λ)ϕl (λ) fkl(λ)dλ < ∞. Denote by LL− 2 (F ) the subspace in L2(F ), generated by functions eitλδk, δk = {δkl}T l=1 , k = 1, T , t ∈ R\[0, L], where δkk = 1, δkl = 0 for k �= l. Any linear estimate ÂL �ξ of the functional AL �ξ based on observations of the process �ξ(t) + �η(t) for t ∈ R\[0, L] is of the form ÂL �ξ = ∞∫ −∞ h(λ) (Zξ(dλ) + Zη(dλ)) = ∞∫ −∞ T∑ k=1 hk(λ)(Zξ k(dλ) + Zη k (dλ)), where Zξ(Δ) = { Zξ k(Δ) }T k=1 and Zη(Δ) = {Zη k (Δ)}T k=1 are orthogonal random measures of the stationary processes �ξ(t) and �η(t) correspondingly, h(λ) = {hk(λ)}T k=1 is the spectral characteristic of the estimate ÂL �ξ. The function h(λ) ∈ LL− 2 (F +G). The value of the mean square error Δ(h; F, G) of the estimate ÂL �ξ is calculated by the formula Δ(h; F, G) = E ∣∣∣AL �ξ − ÂL �ξ ∣∣∣2 = 1 2π ∞∫ −∞ (AL(λ)− h(λ)) F (λ) (AL(λ)− h(λ))∗dλ + 1 2π ∞∫ −∞ h(λ) G(λ) h∗(λ)dλ, where AL(λ) = L∫ 0 �a(t)eitλdt. ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 91 The spectral characteristic h(F, G) of the optimal estimate of the functional AL �ξ in the case of given spectral density matrices F (λ) and G(λ) is deter- mined by the extremum condition Δ(F, G) = Δ(h(F, G); F, G) = (2) = min h∈LL− 2 (F+G) Δ(h; F, G) = min ÂL �ξ E ∣∣∣AL �ξ − ÂL �ξ ∣∣∣2 . The optimal estimate ÂL �ξ is a solution of the extremum problem (2). It is determined by two conditions (see, for example, selected works of A. N. Kol- mogorov (1992)) (3) ÂL �ξ ∈ H [ ξk(t) + ηk(t) , k = 1, T , t ∈ R\[0, L] ] ; (4) AL �ξ − ÂL �ξ⊥H [ ξk(t) + ηk(t) , k = 1, T , t ∈ R\[0, L] ] , where H [ ξk(t) + ηk(t), k = 1, T , t ∈ R\[0, L] ] is a subspace generated by the random variables { ξk(t) + ηk(t), k = 1, T , t ∈ R\[0, L] } in the Hilbert space H of the second order random variables with zero mean value. These conditions give a possibility to find the spectral characteristic h(F, G) and the mean-square error Δ(F, G) of the optimal estimate of the functional AL �ξ under the condition that the spectral density matrices F (λ), G(λ) are known and satisfy the minimality condition (1). In this case h(F, G) = (AL(λ) F (λ) − CL(λ)) (F (λ) + G(λ))−1 = (5) = AL(λ) − (AL(λ) G(λ) + CL(λ)) (F (λ) + G(λ))−1, Δ(F, G) = 1 2π ∞∫ −∞ (AL(λ) G(λ) + CL(λ)) (F (λ) + G(λ))−1 F (λ)× ×(F (λ) + G(λ))−1(AL(λ) G(λ) + CL(λ))∗dλ+ + 1 2π ∞∫ −∞ (AL(λ) F (λ) − CL(λ))(F (λ) + G(λ))−1G(λ) (F (λ) + G(λ))−1× (6) ×(AL(λ) G(λ) + CL(λ))∗dλ = 〈BLc , c〉 + 〈RLa , a〉 , where CL(λ) = L∫ 0 �c(t)eitλdt, �c(t) = (B−1 L DLa)(t), 0 ≤ t ≤ L, 〈BLc, c〉 = L∫ 0 T∑ k=1 (BLc)k(t) ck(t) dt, 92 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA 〈RLa, a〉 = L∫ 0 T∑ k=1 (RLa)k(t) ak(t)dt, operators BL, DL, RL are determined by the following relations (BLc)(t) = 1 2π L∫ 0 ∞∫ −∞ �c(u)(F (λ) + G(λ))−1ei(u−t)λdλdu, (DLc)(t) = 1 2π L∫ 0 ∞∫ −∞ �c(u)F (λ)(F (λ) + G(λ))−1ei(u−t)λdλdu, (RLc)(t) = 1 2π L∫ 0 ∞∫ −∞ �c(u)F (λ)(F (λ) + G(λ))−1G(λ)ei(u−t)λdλdu, 0 ≤ t ≤ L; From the preceding formulas we can conclude that the following theorem holds true. Theorem 1.1. Let �ξ(t) = {ξk(t)}T k=1 , E�ξ(t) = 0, and �η(t) = {ηk(t)}T k=1 , E�η(t) = 0, be uncorrelated multidimensional stationary stochastic processes with the spectral density matrices F (λ) = {fij(λ)}T i,j=1 and G(λ) = {gij(λ)}T i,j=1 that satisfy the minimality condition (1). The value of the mean-square error Δ(h(F, G), F, G) and the spectral characteristic h(F, G) of the optimal linear estimate of the functional AL �ξ = ∫ L 0 �a(t)�ξ(t)dt which depends on the unknown values of the process �ξ(t) based on observations of the process �ξ(t) + �η(t) for t ∈ R\[0, L] are calculated by formulas (5), (6). Corollary 1.1. Let �ξ(t) = {ξk(t)}T k=1 be a multidimensional stationary stochastic process with the spectral density matrix F (λ) = {fkl(λ)}T k,l=1 , that satisfies the minimality condition ∞∫ −∞ b(λ)(F (λ))−1b∗(λ)dλ < ∞ for a nontrivial vector function of the exponential type b(λ) = ∫ L 0 �α(t)eitλdt. The value of the mean-square error Δ(F ) and the spectral characteristic h(F ) of the optimal linear estimate of the functional AL �ξ = ∫ L 0 �a(t)�ξ(t)dt which depends on the unknown values of the process �ξ(t) based on observa- tions of the process �ξ(t) for t ∈ R\[0, L] can be calculated by the formulas (7) h(F ) = AL(λ) − CL(λ) (F (λ))−1, ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 93 (8) Δ(F ) = 1 2π ∞∫ −∞ CL(λ) (F (λ))−1(CL(λ))∗dλ = 〈BLa, a〉 = 〈c, a, 〉 where CL(λ) = L∫ 0 �c(t)eitλdt, �c(t) = (B−1 L a)(t), 0 ≤ t ≤ L, 〈c, a〉 = L∫ 0 T∑ k=1 ck(t) ak(t)dt, (BLa)(t) = 1 2π L∫ 0 ∞∫ −∞ �a(u)(F (λ))−1ei(u−t)λdλdu, 0 ≤ t ≤ L. Example 1. Consider the problem of estimation of the value of the func- tional A1 �ζ = ∫ 1 0 �a(t)�ζ(t)dt based on observations of the process �ζ(t) = (ζ1(t), ζ2(t)), t ∈ R\[0, 1], where ζ1(t) = ξ(t) is a stationary stochastic process with the spectral density f(λ), and ζ2(t) = ξ(t) + η(t), where η(t) is an uncorrelated with ξ(t) stationary stochastic process with the spectral density g(λ). In this case F (λ) = ( f(λ) f(λ) f(λ) f(λ) + g(λ) ) . The determinant D = \|F (λ)\| = f(λ) g(λ), and the inverse matrix F (λ)−1 = ( f(λ)+g(λ) f(λ) g(λ) −1 g(λ) −1 g(λ) 1 g(λ) ) . Let f(λ) = P1 λ2+ α2 1 , g(λ) = P2 λ2+α2 2 , �a(t) = (1, 1). Then A1(λ) = eiλ−1 iλ �a(t), and the function �c(t) = (c1(t), c2(t)) are determined by the equation (B1c)(t) = �a(t), where (B1c)(t) = 1 2π 1∫ 0 ∞∫ −∞ �c(u)(F (λ))−1ei(u−t)λdλdu. We get the following system of integral equations with respect to c1(t) and c2(t) 1 2π 1∫ 0 ∞∫ −∞ c1(u) α2 1 + λ2 P1 ei(u−t)λdλdu = 2, 94 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA 1 2π 1∫ 0 ∞∫ −∞ (c2(u) − c1(u)) α2 2 + λ2 P1 ei(u−t)λdλdu = 1. Transform these integral equations to the differential ones and get c1(t) = P1x(t), x′′(t) − α1x(t) + 2 = 0, x(0) = x(1) = 0, c2(t) = c1(t) + P2y(t), y′′(t) − α2y(t) + 1 = 0, y(0) = y(1) = 0. Solutions to these equations are of the form c1(t) = 2P1 α2 1 ( eα1 − 1 e−α1 − eα1 e−α1t − e−α1 − 1 e−α1 − eα1 eα1t + 1 ) , c2(t) = c1(t) + P2 α2 2 ( eα2 − 1 e−α2 − eα2 e−α2t − e−α2 − 1 e−α2 − eα2 eα2t + 1 ) . The spectral characteristic of the optimal estimate of the value A1 �ζ is of the form h(F ) = A1(λ) − C1(λ) F (λ)−1 = (h1(λ), h2(λ)), where h1(λ) = (h1 − h2) + (h1 − h2)e iλ, h2(λ) = h2 + h2e iλ, h1 = 2 α1(e−α1 − eα1) (2 − eα1 − e−α1), h2 = 1 α2(e−α2 − eα2) (2 − eα2 − e−α2). The optimal estimate of the value A1 �ζ is of the form Â1 �ζ = (h1 − h2)ζ1(0) + h2ζ2(0) + (h1 − h2)ζ1(1) + h2ζ2(1). The mean-square error of the optimal estimate can be calculated by the formula Δ(F ) = 4P1 α2 1 ( 1 + 2(1 − e−α1)(eα1 − 1) α1(e−α1 − eα1) ) + P2 α2 2 ( 1 + 2(1 − e−α2)(eα2 − 1) α2(e−α2 − eα2) ) . Example 2. Consider the problem of estimation of the value A1 �ζ =∫ 1 0 �a(t)�ζ(t)dt, where �a(t) = (1, 1 − t). In this case c1(t) = P1 ( Ae−α1t + Beα1t − 1 α2 1 t + 2 α2 1 ) , A = 1 α2 1 2eα1 − 1 e−α1 − eα1 , B = − 1 α2 1 2e−α1 − 1 e−α1 − eα1 ; c2(t) = c1(t) + P2 ( Ce−α2t + Deα2t − 1 α2 2 t + 1 α2 2 ) , C = 1 α2 2 eα2 e−α2 − eα2 , D = − 1 α2 2 e−α2 e−α2 − eα2 ; A1(λ) = ( eiλ − 1 iλ , 1 λ2 − 1 iλ − eiλ λ2 ) . The spectral characteristic of the optimal estimate is of the form h(F ) = (h1(λ),h2(λ)) , h1(λ) = (h1 −h2)+ (h3 −h4)e iλ, h2(λ) = h2 +h4e iλ, ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 95 where h1 = 2 − 2(e−α1 + eα1) α1(e−α1 − eα1) − 1 α2 1 , h2 = − 1 α2 e−α2 + eα2 e−α2 − eα2 − 1 α2 2 , h3 = 4 − (e−α1 + eα1) α1(e−α1 − eα1) + 1 α2 1 , h4 = 2 α2(e−α2 − eα2) + 1 α2 2 . The optimal estimate of the value A1 �ζ is of the form Â1 �ζ = (h1 − h2)ζ1(0) + (h3 − h4)ζ1(1) + h2ζ2(0) + h4ζ2(1). The mean-square error of the optimal estimate can be calculated by the formula Δ(F ) = 2P1 ( 1 3α2 1 + A ( 1 α1 + e−α1 α2 1 − 1 α2 1 ) + B ( − 1 α1 + eα1 α2 1 − 1 α2 1 )) + +P2 ( − 1 6α2 2 + C ( 1 α2 + e−α2 α2 2 − 1 α2 2 ) + D ( − 1 α2 + eα2 α2 2 − 1 α2 2 )) . Example 3. Consider the problem of estimation of the value A1 �ζ =∫ 1 0 �a(t)�ζ(t)dt, where �a(t) = (1, eβt). In this case A1(λ) = ( eiλ − 1 iλ , eiλ+β − 1 iλ + β ) , c1(t) = P1 ( Ae−α1t + Beα1t + eβt α2 1 − β2 + 1 α2 1 ) , c2(t) = c1(t) + P2 ( Ce−α2t + Deα2t + eβt α2 2 − β2 ) , A = 1 e−α1 − eα1 ( eα1 − eβ α2 1 − β2 + eα1 − 1 α2 1 ) , B = − 1 e−α1 − eα1 ( e−α1 − eβ α2 1 − β2 + e−α1 − 1 α2 1 ) , C = 1 e−α2 − eα2 eα2 − eβ α2 2 − β2 , D = − 1 e−α2 − eα2 e−α2 − eβ α2 2 − β2 . The spectral characteristic of the optimal estimate is of the form h(F ) = (h1(λ) , h2(λ)), h1(λ) = (h1−h2)+(h3−h4)e iλ, h2(λ) = h2+h4e iλ, where h1 = α1 α2 1 − β2 2eβ − e−α1 − eα1 e−α1 − eα1 + 2 − e−α1 − eα1 α1(e−α1 − eα1) + β α2 1 − β2 , h3 = α1 α2 1 − β2 2 − eβ(e−α1 + eα1) e−α1 − eα1 + 2 − e−α1 − eα1 α1(e−α1 − eα1) − βeβ α2 1 − β2 , h2 = α2(2e β − e−α2 − eα2) (e−α2 − eα2)(α2 2 − β2) + β α2 2 − β2 , 96 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA h4 = α2(2 − eβe−α2 − eα2) (e−α2 − eα2)(α2 2 − β2) − β α2 2 − β2 eβ. The mean-square error of the optimal estimate can be calculated by the formula Δ(F ) = 2P1 ( 1 α2 1 − A e−α1 − 1 α1 + B eα1 − 1 α1 + eβ − 1 β(α2 1 − β2) ) + +P2 ( C eβ−α2 − 1 β − α2 + D eβ+α2 − 1 β + α2 + e2β − 1 2β(α2 2 − β2) ) . 3. Minimax-robust method of interpolation Formulas (5)-(8) may be used to determine the mean-square error and the spectral characteristic of the optimal linear estimate of the functional AL �ξ when the spectral density matrices F (λ) and G(λ) of multidimensional sta- tionary stochastic processes �ξ(t) and �η(t) are known. In the case where the spectral density matrices are unknown, but a set D = DF × DG of ad- missible spectral density matrices is given, the minimax-robust method of estimation of the unknown values of the functional AL �ξ is reasonable (see, for example, the survey article by S. A. Kassam and H. V. Poor (1985)). By means of this method it is possible to determine the estimate that minimizes the mean-square error for all spectral density matrices F (λ) and G(λ) from the class D = DF × DG simultaneously. Definition 3.1. For a given class of spectral density matrices D = DF ×DG spectral density matrices F 0 (λ) ∈ DF , G0 (λ) ∈ DG are called the least favorable for the optimal linear interpolation of the functional AL �ξ if the following relation holds true Δ ( F 0, G0 ) = Δ ( h ( F 0, G0 ) ; F 0, G0 ) = max (F, G)∈D Δ (h (F, G) ; F, G) . Definition 3.2. For a given class of spectral density matrices D = DF × DG the spectral characteristic h0(λ) of the optimal linear estimate of the functional AL �ξ is called the minimax-robust if the conditions h0 (λ) ∈ HD = ∩ (F,G)∈D LL− 2 (F + G) , min h∈HD max (F, G)∈D Δ (h; F, G) = max (F, G)∈D Δ ( h0; F, G ) . are satisfied. Taking into account relations (1)–(8), it is possible to verify the following propositions. Proposition 3.1. The spectral density matrices F 0(λ) ∈ DF , G0(λ) ∈ DG are the least favorable in the class D = DF × DG for the optimal linear interpolation of the functional AL �ξ if the density matrix functions (F 0(λ)+ G0(λ))−1, F 0(λ)(F 0(λ)+G0(λ))−1, F 0(λ)(F 0(λ)+G0(λ))−1G0(λ) determine ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 97 operators B0 L, D0 L, R0 L, which give solutions to the conditional extremum problem (9) max (F, G)∈D (〈 DLa, B−1 L DLa 〉 + 〈RLa, a〉) = 〈 D0 La, (B0 L)−1D0 La 〉 + 〈 R0 La, a 〉 . The minimax-robust spectral characteristic h0 = h(F 0, G0) of the optimal linear estimate of the functional AL �ξ can be calculated by formula (5) if the condition h(F 0, G0) ∈ HD holds true. Proposition 3.2. The spectral density matrix F 0(λ) ∈ DF which satisfies the minimality condition is the least favorable in the class D = DF for the optimal linear interpolation of the functional AL �ξ based on observations of �ξ(t), t ∈ R\[0, L], if the density matrix function (F 0(λ))−1 determine the operator B0 L which gives a solution to the conditional extremum problem (10) max F∈DF 〈 (BL)−1a, a 〉 = 〈 (B0 L)−1a, a 〉 . The minimax-robust spectral characteristic h0 = h(F 0) of the optimal lin- ear estimate of the functional AL �ξ can be calculated by formula (7) if the condition h(F 0) ∈ HD holds true. The least favorable spectral density matrices F 0(λ) ∈ D, G0(λ) ∈ DG and the minimax-robust spectral characteristic h0 = h (F 0, G0) ∈ HD form a saddle point of the function Δ(h; F, G) on the set HD × D. The saddle point inequalities Δ ( h0; F, G ) ≤ Δ ( h0; F 0, G0 ) ≤ Δ ( h; F 0, G0 ) , ∀h ∈ HD, ∀F ∈ DF , ∀G ∈ DG hold true if h0 = h(F 0, G0) ∈ HD and (F 0, G0) give a solution to the conditional extremum problem (11) sup (F,G)∈D Δ ( h ( F 0, G0 ) ; F, G ) = Δ ( h ( F 0, G0 ) ; F 0, G0 ) , where Δ ( h ( F 0, G0 ) ; F, G ) = = 1 2π ∞∫ −∞ (AL(λ) G0(λ) + C0 L(λ)) (F 0(λ) + G0(λ))−1 F (λ)× ×(F 0(λ) + G0(λ))−1(AL(λ) G0(λ) + C0 L(λ))∗dλ+ + 1 2π ∞∫ −∞ (AL(λ) F 0(λ) − C0 L(λ))(F 0(λ) + G0(λ))−1G(λ)× ×(F 0(λ) + G0(λ))−1(AL(λ) G0(λ) + C0 L(λ))∗dλ. 98 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA This conditional extremum problem is equivalent to the unconditional ex- tremum problem (12) ΔD(F, G) = −Δ(h(F 0, G0); F, G) + δ((F, G)\|D) → inf, where δ((F, G)\|D) is the indicator function of the set D = DF ×DG. A solu- tion to this problem is determined by the condition 0 ∈ ∂ΔD(F 0, G0), where ∂ΔD(F 0, G0) is the subdifferential of the convex functional ∂ΔD(F, G) at the point (F 0, G0) (see, for example, B. N. Pshenichnyi (1982)). The following propositions holds true. Proposition 3.3. Let (F 0, G0) be a solution to the conditional extremum problem (11). The spectral density matrices F 0(λ) ∈ DF , G0(λ) ∈ DG are the least favorable in the class D = DF ×DG and the spectral characteristic h0 = h(F 0, G0) is minimax-robust for the optimal linear interpolation of the functional AL �ξ if the condition h(F 0, G0) ∈ HD holds true. Proposition 3.4. The spectral density matrix F 0(λ) ∈ DF which satisfies the minimality condition is the least favorable in the class D = DF for the optimal linear interpolation of the functional AL �ξ based on observations of �ξ(t), t ∈ R\[0, L], if the density matrix function F 0(λ) gives a solution to the conditional extremum problem (13) sup F∈DF Δ ( h ( F 0 ) ; F ) = Δ ( h ( F 0 ) ; F 0 ) , where Δ ( h ( F 0 ) ; F ) = 1 2π ∞∫ −∞ C0 L(λ)(F 0(λ))−1F (λ)(F 0(λ))−1(C0 L(λ))∗dλ. The spectral characteristic h0 = h(F 0) is minimax-robust for the optimal linear interpolation of the functional AL �ξ if the condition h(F 0) ∈ HD holds true. 4. Least favorable spectral densities in the class D0 F × D0 G Consider the problem of minimax estimation of the functional AL �ξ based on observations �ξ(t) + �η(t), t ∈ R\[0, L] under the condition that spectral density matrices F (λ), G(λ) of the multidimentional stationary processes �ξ(t), �η(t) are from the set of spectral density matrices D0 F × D0 G, where D0 F = ⎧⎨ ⎩F (λ) ∣∣∣∣∣∣ 1 2π ∞∫ −∞ F (λ)dλ = P1 ⎫⎬ ⎭ , D0 G = ⎧⎨ ⎩G(λ) ∣∣∣∣∣∣ 1 2π ∞∫ −∞ G(λ)dλ = P2 ⎫⎬ ⎭ . ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 99 With the help of the Lagrange multipliers method we can find the follow- ing relations that determine the least favorable spectral density matrices (F 0(λ), G0(λ)) ∈ D0 F × D0 G (14) AL(λ)G0(λ) + C0 L(λ) = �α · (F 0(λ) + G0(λ)), (15) AL(λ)F 0(λ) − C0 L(λ) = �β · (F 0(λ) + G0(λ)), where �α = (α1, . . . , αT ), �β = (β1, . . . , βT ) are the Lagrange multipliers. It follows from these relations that the the following theorems hold true. Theorem 4.1. Let the spectral density matrices F 0(λ) ∈ D0 F , G0(λ) ∈ D0 G satisfy the minimality condition (1). These spectral density matrices F 0(λ), G0(λ) are the least favorable in the class D = D0 F × D0 G for the optimal linear interpolation of the functional AL �ξ if F 0(λ), G0(λ) are solutions to the equations (14), (15) and determine a solution to the extremum problem (9). The spectral characteristic h(F 0, G0) calculated by the formula (5) is minimax-robust for the optimal linear interpolation of the functional AL �ξ. Theorem 4.2. Let the spectral density matrix F (λ) be know and let spectral density matrices F (λ), G0(λ) ∈ D0 G satisfy the minimality condition (1). The spectral density matrix G0(λ) is the least favorable in the class D0 G for the optimal linear interpolation of the functional AL �ξ if G0(λ) = max { 0, �α−1 · (AL(λ)F (λ) − C0 L(λ)) − F (λ) } , where �α−1 = ( α1 D , . . . , αT D )� , D = \|α1\|2 + . . . + \|αT \|2 , and F (λ), G0(λ) determine a solution to the extremum problem (9). The spectral characteristic h(F, G0) calculated by the formula (5) is minimax- robust for the optimal linear interpolation of the functional AL �ξ. Theorem 4.3. Let the spectral density matrix F 0(λ) ∈ D0 F satisfies the minimality condition (1). The spectral density matrix F 0(λ) is the least favorable in the class D = D0 F for the optimal linear interpolation of the functional AL �ξ based on observations of �ξ(t) for t ∈ R\[0, L] if �α · F 0(λ) = C0 L(λ) and F 0(λ) determine a solution to the extremum problem (10). The spectral characteristic h0 = h(F 0) calculated by the formula (7) is minimax- robust for the optimal linear interpolation of the functional AL �ξ. 5. Least favorable spectral densities in the class DU V × Dε Consider the problem of minimax estimation of the functional AL �ξ based on observations �ξ(t) + �η(t), t ∈ R\[0, L] under the condition that spectral density matrices F (λ), G(λ) of the multidimensional stationary processes 100 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA �ξ(t), �η(t) are from the set of spectral density matrices DU V × Dε, where DU V = ⎧⎨ ⎩F (λ) ∣∣∣∣∣∣V (λ) ≤ F (λ) ≤ U(λ), 1 2π ∞∫ −∞ F (λ)dλ = P1 ⎫⎬ ⎭ , Dε = ⎧⎨ ⎩G(λ) ∣∣∣∣∣∣G(λ) = (1 − ε)G1(λ) + εW (λ), 1 2π ∞∫ −∞ G(λ)dλ = P2 ⎫⎬ ⎭ , where V (λ), U(λ), G1(λ) are given fixed spectral density matrices, W (λ) ia an unknown spectral density matrix, and expression B(λ) ≥ D(λ) means that B(λ) − D(λ) ≥ 0 (positive definite matrix function). The set DU V describes the ‘band’ model of stochastic processes while the set Dε describes the ‘ε-contaminated’ model of stochastic processes. For the set DU V × Dε from the condition 0 ∈ ∂ΔD(F 0, G0) we can get the following relations which determine the least favorable spectral density matrices (16) �a0(λ)�a0(λ)∗ = �α · �α∗ + Γ1(λ) + Γ2(λ); (17) �b0(λ)�b0(λ)∗ = �β · �β∗ + Γ3(λ), where �a0(λ) = ( (AL(λ)G0(λ) + C0 L(λ))(F 0(λ) + G0(λ))−1 )T , �b0(λ) = ( (AL(λ)F 0(λ) − C0 L(λ))(F 0(λ) + G0(λ))−1 )T . The coefficients �α = (α1, . . . , αT )T , �β = (β1, . . . βT )T are determined by the conditions (18) 1 2π ∞∫ −∞ F 0(λ)dλ = P1, 1 2π ∞∫ −∞ G0(λ)dλ = P2. The matrix functions Γ1(λ) ≥ 0, Γ2(λ) ≥ 0, Γ3(λ) ≥ 0 are determined by the conditions (19) V (λ) ≤ F 0(λ) ≤ U(λ), G0(λ) = (1 − ε)G1(λ) + εW (λ), (20) Γ1(λ) = 0 if F 0(λ) ≥ V (λ), Γ2(λ) = 0 if F 0(λ) ≤ U(λ), (21) Γ3(λ) = 0 if G0(λ) ≥ (1 − ε)G1(λ). It follows from these relations that the the following theorems hold true. Theorem 5.1. Let the spectral density matrices F 0(λ) ∈ DU V , G0(λ) ∈ Dε satisfy the minimality condition (1). These spectral density matrices F 0(λ), G0(λ) are the least favorable in the class DU V × Dε for the optimal linear interpolation of the functional AL �ξ if they satisfy conditions (16) – (21) and determine a solution to the extremum problem (9). The spectral ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 101 characteristic h(F 0, G0) calculated by the formula (5) is minimax-robust for the optimal linear interpolation of the functional AL �ξ. Theorem 5.2. Let the spectral density matrix F (λ) be known and let spec- tral density matrices F (λ), G0(λ) ∈ Dε satisfy the minimality condition (1). The spectral density matrix G0(λ) is the least favorable in the class Dε for the optimal linear interpolation of the functional AL �ξ if G0(λ) = max { (1 − ε)G1(λ), �α−1(AL(λ)F (λ) − C0 L(λ)) − G(λ) } and (F (λ), G0(λ)) determine a solution to the extremum problem (9). The spectral characteristic h(F, G0) calculated by the formula (5) is minimax- robust for the optimal linear interpolation of the functional AL �ξ. Theorem 5.3. Let the spectral density matrix F 0(λ) ∈ DU V satisfies the minimality condition (1). This spectral density matrix F 0(λ) is the least favorable in the class D = DU V for the optimal linear interpolation of the functional AL �ξ based on observations of �ξ(t) for t ∈ R\[0, L] if F 0(λ) = max { V (λ), min { U(λ), �α−1C0 L(λ) }} and F 0(λ) determine a solution to the extremum problem (10). The spectral characteristic h0 = h(F 0) calculated by the formula (7) is minimax-robust for the optimal linear interpolation of the functional AL �ξ. 6. Hilbert space projection method of extrapolation Let the vector function �a(t) which determines the functional A�ξ satisfies conditions: (22) ∫ ∞ 0 T∑ k=1 \|ak(t)\| dt < ∞, ∫ ∞ 0 t T∑ k=1 \|ak(t)\|2dt < ∞. Under these conditions the functional A�ξ has the second moment and the operator A defined below is compact. Let the spectral density matrices F (λ) = {fkl(λ)}T k,l=1 and G(λ) = {gkl(λ)}T k,l=1 of uncorrelated multidimensional stationary stochastic processes �ξ(t) = {ξk(t)}T k=1 , �η(t) = {ηk(t)}T k=1 satisfy the minimality condition (1), where b(λ) = ∫∞ 0 �α(t)eitλdt, �α(t) = {αk(t)}T k=1, is a nontrivial vector function of the exponential type. Denote by L− 2 (F ) the subspace in L2(F ), generated by functions eitλδk, t < 0, δk = {δkl}T l=1 , k = 1, . . . , T , where δkk = 1, δkl = 0 for k �= l. Any linear linear extrapolation Â�ξ of the functional A�ξ based on observations of the process �ξ(t) + �η(t) for t < 0 is of the form Â�ξ = ∞∫ −∞ h(λ) (Zξ(dλ) + Zη(dλ)) = ∞∫ −∞ T∑ k=1 hk(λ)(Zξ k(dλ) + Zη k (dλ)), 102 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA where h(λ) = {hk(λ)}T k=1 is the spectral characteristic of the linear extrap- olation Â�ξ. The function h(λ) ∈ L− 2 (F + G). The value of the mean square error Δ(h; F, G) of the linear extrapolation Â�ξ is calculated by the formula Δ(h; F, G) = E ∣∣∣A�ξ − Â�ξ ∣∣∣2 = 1 2π ∞∫ −∞ (A(λ) − h(λ)) F (λ) (A(λ)− h(λ))∗dλ + 1 2π ∞∫ −∞ h(λ) G(λ) h∗(λ)dλ, where A(λ) = ∞∫ 0 �a(t)eitλdt. The spectral characteristic h(F, G) of the optimal linear linear extrapolation of A�ξ minimizes the mean square error Δ(F, G) = Δ(h(F, G); F, G) (23) = min h∈L− 2 (F+G) Δ(h; F, G) = min Â�ξ E ∣∣∣A�ξ − Â�ξ ∣∣∣2 . With the help of the Hilbert space projection method proposed by A. N. Kol- mogorov we can find a solution of the optimization problem (23): h(F, G) = (A(λ) F (λ) − C(λ)) (F (λ) + G(λ))−1 = (24) = A(λ) − (A(λ) G(λ) + C(λ)) (F (λ) + G(λ))−1, Δ(F, G) = 1 2π ∞∫ −∞ (A(λ) G(λ) + C(λ)) (F (λ) + G(λ))−1 F (λ)× ×(F (λ) + G(λ))−1(A(λ) G(λ) + C(λ))∗dλ+ + 1 2π ∞∫ −∞ (A(λ) F (λ) − C(λ))(F (λ) + G(λ))−1G(λ) (F (λ) + G(λ))−1× (25) ×(A(λ) G(λ) + C(λ))∗dλ = 〈Bc , c〉 + 〈Ra , a〉 , where C(λ) = ∞∫ 0 �c(t)eitλdt, �c(t) = (B−1Da)(t), 〈Bc, c〉 = ∞∫ 0 n∑ k=1 (Bc)k(t) ck(t) dt, ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 103 〈Ra, a〉 = ∞∫ 0 n∑ k=1 (Ra)k(t) ak(t)dt, operators B,D,R are determined by the following relations (Bc)(t) = 1 2π ∞∫ 0 ∞∫ −∞ �c(u)(F (λ) + G(λ))−1ei(u−t)λdλdu, (Dc)(t) = 1 2π ∞∫ 0 ∞∫ −∞ �c(u)F (λ)(F (λ) + G(λ))−1ei(u−t)λdλdu, (Rc)(t) = 1 2π ∞∫ 0 ∞∫ −∞ �c(u)F (λ)(F (λ) + G(λ))−1G(λ)ei(u−t)λdλdu. From the preceding formulas we can conclude that the following theorem holds true. Theorem 6.1. Let �ξ(t) = {ξk(t)}T k=1 and �η(t) = {ηk(t)}T k=1 be uncorrelated multidimensional stationary stochastic processes with the spectral density matrices F (λ) = {fij(λ)}T i,j=1 and G(λ) = {gij(λ)}T i,j=1 that satisfy the min- imality condition (1). Let condition (22) be satisfied. The value of the mean-square error Δ(h(F, G), F, G) and the spectral characteristic h(F, G) of the optimal linear extrapolation of the functional A�ξ = ∫∞ 0 �a(t)�ξ(t)dt which depends on the unknown values of the process �ξ(t) t ≥ 0, based on observations of the process �ξ(t) + �η(t) for t < 0 are calculated by formulas (24), (25). Corollary 6.1. Let �ξ(t) = {ξk(t)}T k=1 be a multidimensional stationary stochastic process with the spectral density matrix F (λ) = {fkl(λ)}T k,l=1 , that satisfies the minimality condition ∞∫ −∞ b(λ)(F (λ))−1b∗(λ)dλ < ∞ for a nontrivial vector function of the exponential type b(λ) = ∫∞ 0 �α(t)eitλdt. Let condition (22) be satisfied. The value of the mean-square error Δ(F ) and the spectral characteristic h(F ) of the optimal linear extrapolation of the functional A�ξ = ∫∞ 0 �a(t)�ξ(t)dt based on observations of the process �ξ(t) for t < 0 are calculated by the formulas (26) h(F ) = A(λ) − C(λ) (F (λ))−1, (27) Δ(F ) = 1 2π ∞∫ −∞ C(λ) (F (λ))−1(C(λ))∗dλ = 〈Ba, a〉 = 〈c, a〉 104 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA where �c(t) = (B−1a)(t), (Ba)(t) = 1 2π ∞∫ 0 ∞∫ −∞ �a(u)(F (λ))−1ei(u−t)λdλdu. Let the process �ξ(t) admits the canonical moving average representation (28) �ξ(t) = ∫ t −∞ d(t − u) d�ε(u), where d(s) = {dij(s)}j=1,m i=1,T is a matrix function and �ε(u) = {εk(u)}m k=1 is a multidimensional stationary stochastic process with uncorrelated incre- ments. In this case the spectral density matrix F (λ) = {fij(λ)}n i,j=1 of the process �ξ(t) admits the canonical factorization: (29) F (λ) = ϕ(λ) ϕ∗(λ), ϕ(λ) = ∫ ∞ 0 d(u)e−iuλdλ. If the process �ξ(t) admits the canonical moving average representation (28), then the optimal linear extrapolation of the functional A�ξ = ∫∞ 0 �a(t)�ξ(t)dt based on observations of the process �ξ(t) for t < 0 is determined by the spectral characteristic h(F ) ∈ L− 2 (F ) that minimizes the mean square error (30) Δ(h(F ), F ) = min h∈L− 2 (F ) Δ(h, F ) = ‖Ad‖2 , where (Ad)(t) = ∫ ∞ 0 �a(t + u)d(u)du, ‖Ad‖2 = ∫ ∞ 0 m∑ k=1 \|(Ad)k(t)\|2 dt. Note, that ‖Ad‖2 < ∞ under condition (22). The spectral characteristic h(F ) is calculated by the formula (31) h(F ) = A(λ) − r(λ)ψ (λ), r(λ) = ∫ ∞ 0 (Ad)(t)eitλdt. Here ψ (λ) = {ψij(λ)}j=1,T i=1,m is a matrix function which satisfies the equation ψ (λ)ϕ(λ) = Im, where Im is the identity matrix of order m. For the functional AL �ξ = ∫ L 0 �a(t)�ξ(t)d(t) the value of the mean square error and the spectral characteristics of the optimal linear extrapolation are determined by the following formulas (32) ΔL(h(F ), F ) = ‖ALd‖2 , (33) h(F ) = AL(λ) − rL(λ)ψ (λ), ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 105 where ALd(t) = ∫ L−t 0 �a(t + u)d(u)du, ‖AT d‖2 = ∫ L 0 m∑ k=1 \|(ALd)k(t)\|2dt, AL(λ) = ∫ L 0 �a(t)eitλdt, rL(λ) = ∫ L 0 (ALd)(t)eitλdt. As a corollary we can get the following formulas for calculation the mean square error of the optimal linear extrapolation ξ̂k(L) of the unknown values ξk(L), k = 1, . . . , T : (34) E ∣∣∣ξk(L) − ξ̂k(L) ∣∣∣2 = ∫ L 0 m∑ l=1 \|dkl(t)\|2dt. The following theorem holds true. Theorem 6.2. Let �ξ(t) = {ξk(u)}n k=1 be a stationary stochastic process that admits the canonical moving average representation (28) with the spectral density matrix F (λ) that admits the canonical factorization (29) and let condition(22) be satisfied. The value of the mean-square error Δ(h(F ), F ) of the optimal linear extrapolation of the functional A�ξ from observations of the process �ξ(t) for t < 0 is calculated by formula (30) (by formula (32) if the functional AL �ξ is estimated). The spectral characteristics h(F ) of the optimal linear extrapolation can be calculated by formula (31) (by formula (33) if the functional AL �ξ is estimated). 7. Minimax-robust method of extrapolation Taking into account relations (22)–(27), we can verify the following propo- sitions. Proposition 7.1. The spectral density matrices F 0(λ) ∈ DF , G0(λ) ∈ DG are the least favorable in the class D = DF × DG for the optimal linear extrapolation of the functional A�ξ if the density matrix functions (F 0(λ) + G0(λ))−1, F 0(λ)(F 0(λ)+G0(λ))−1, F 0(λ)(F 0(λ)+G0(λ))−1G0(λ) determine operators B0, D0, R0, which give solutions to the conditional extremum problem (35) max (F, G)∈D (〈 Da,B−1Da 〉 + 〈Ra, a〉) = 〈 D0a, (B0)−1D0a 〉 + 〈 R0a, a 〉 . The minimax-robust spectral characteristic h0 = h(F 0, G0) of the optimal linear extrapolation of the functional A�ξ is calculated by formula (24) if the condition h(F 0, G0) ∈ HD holds true. Proposition 7.2. The spectral density matrix F 0(λ) ∈ DF which satisfies the minimality condition is the least favorable in the class D = DF for the optimal linear extrapolation of the functional A�ξ based on observations of 106 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA �ξ(t), t < 0, if the density matrix function (F 0(λ))−1 determine the operator B0 which gives a solution to the conditional extremum problem (36) max F∈DF 〈 (B)−1a, a 〉 = 〈 (B0)−1a, a 〉 . The minimax-robust spectral characteristic h0 = h(F 0) of the optimal lin- ear extrapolation of the functional A�ξ is calculated by formula (26) if the condition h(F 0) ∈ HD holds true. The least favorable spectral density matrices F 0(λ) ∈ D, G0(λ) ∈ DG and the minimax-robust spectral characteristic h0 = h (F 0, G0) ∈ HD form a saddle point of the function Δ(h; F, G) on the set HD × D. The saddle point inequalities hold true if h0 = h(F 0, G0) ∈ HD and (F 0, G0) gives a solution to the conditional extremum problem (37) sup (F,G)∈D Δ ( h ( F 0, G0 ) ; F, G ) = Δ ( h ( F 0, G0 ) ; F 0, G0 ) , where Δ ( h ( F 0, G0 ) ; F, G ) = = 1 2π ∞∫ −∞ (A(λ) G0(λ) + C0(λ)) (F 0(λ) + G0(λ))−1 F (λ)× ×(F 0(λ) + G0(λ))−1(A(λ) G0(λ) + C0(λ))∗dλ+ + 1 2π ∞∫ −∞ (A(λ) F 0(λ) − C0(λ))(F 0(λ) + G0(λ))−1G(λ)× ×(F 0(λ) + G0(λ))−1(A(λ) G0(λ) + C0(λ))∗dλ. This conditional extremum problem is equivalent to the unconditional ex- tremum problem (38) ΔD(F, G) = −Δ(h(F 0, G0); F, G) + δ((F, G)\|D) → inf, where δ((F, G)\|D) is the indicator function of the set D = DF × DG. The following propositions hold true. Proposition 7.3. Let (F 0, G0) be a solution to the conditional extremum problem (37). The spectral density matrices F 0(λ) ∈ DF , G0(λ) ∈ DG are the least favorable in the class D = DF ×DG and the spectral characteristic h0 = h(F 0, G0) is minimax-robust for the optimal linear extrapolation of the functional A�ξ if the condition h(F 0, G0) ∈ HD holds true. Proposition 7.4. The spectral density matrix F 0(λ) ∈ DF is the least favorable in the class DF for the optimal linear extrapolation of the func- tional A�ξ based on observations of �ξ(t), t < 0, if F 0(λ) admits the canonical factorization (39) F 0(λ) = [∫ ∞ 0 d0(t)e−itλdt ] · [∫ ∞ 0 d0(t)e−itλdt ]∗ , ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 107 where d0(t) is a solution to the conditional extremum problem (40) ‖Ad‖2 → max, F (λ) = [∫ ∞ 0 d(t)e−itλdt ] · [∫ ∞ 0 d(t)e−itλdt ]∗ ∈ D. The process �ξ(t) in this case admits the canonical moving average represen- tation (41) �ξ(t) = ∫ t −∞ d0(t − u)d�ε(u). The minimax spectral characteristic h0 = h(F 0) is calculated by the formula (31) under the condition h(F 0) ∈ HD. Proposition 7.5. The spectral density matrix F 0(λ) ∈ DF is the least fa- vorable in the class DF for the optimal linear extrapolation of the functional AL �ξ based on observations of �ξ(t), t < 0, if F 0(λ) admits the canonical factorization (42) F 0(λ) = [∫ L 0 d0(t)e−itλdt ] · [∫ L 0 d0(t)e−itλdt ]∗ , where d0(t), 0 ≤ t ≤ L, is a solution to the conditional extremum problem (43) ‖ALd‖2 → max, F (λ) = [∫ L 0 d(t)e−itλdt ] · [∫ L 0 d(t)e−itλdt ]∗ ∈ D. The process �ξ(t) in this case admits the canonical moving average represen- tation (44) �ξ(t) = ∫ t t−L d0(t − u)d�ε(u). The minimax spectral characteristic h0 = h(F 0) is calculated by the formula (33) under the condition h(F 0) ∈ HD. 8. Least favorable spectral densities in the class DU V × Dε Consider the problem of minimax extrapolation of the functional A�ξ based on observations �ξ(t) + �η(t), t < 0, under the condition that spectral density matrices F (λ), G(λ) of the multidimensional stationary processes �ξ(t), �η(t) are from the set of spectral density matrices DU V × Dε, where DU V = ⎧⎨ ⎩F (λ) ∣∣∣∣∣∣V (λ) ≤ F (λ) ≤ U(λ), 1 2π ∞∫ −∞ F (λ)dλ = P1 ⎫⎬ ⎭ , Dε = ⎧⎨ ⎩G(λ) ∣∣∣∣∣∣G(λ) = (1 − ε)G1(λ) + εW (λ), 1 2π ∞∫ −∞ G(λ)dλ = P2 ⎫⎬ ⎭ , where V (λ), U(λ), G1(λ) are given fixed spectral density matrices, W (λ) ia an unknown spectral density matrix, and expression B(λ) ≥ D(λ) means that B(λ) − D(λ) ≥ 0 (positive definite matrix function). For the set 108 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA DU V × Dε from the condition 0 ∈ ∂ΔD(F 0, G0) we can get the following relations which determine the least favorable spectral density matrices (45) �a0(λ)�a0(λ)∗ = �α · �α∗ + Γ1(λ) + Γ2(λ); (46) �b0(λ)�b0(λ)∗ = �β · �β∗ + Γ3(λ), where �a0(λ) = ( (A(λ)G0(λ) + C0(λ))(F 0(λ) + G0(λ))−1 )T , �b0(λ) = ( (A(λ)F 0(λ) − C0(λ))(F 0(λ) + G0(λ))−1 )T . The coefficients �α = (α1, . . . , αT )T , �β = (β1, . . . βT )T are determined by the conditions (47) 1 2π ∞∫ −∞ F 0(λ)dλ = P1, 1 2π ∞∫ −∞ G0(λ)dλ = P2. The matrix functions Γ1(λ) ≥ 0, Γ2(λ) ≥ 0, Γ3(λ) ≥ 0 are determined by the conditions (48) V (λ) ≤ F 0(λ) ≤ U(λ), G0(λ) = (1 − ε)G1(λ) + εW (λ), (49) Γ1(λ) = 0 if F 0(λ) ≥ V (λ), Γ2(λ) = 0 if F 0(λ) ≤ U(λ), (50) Γ3(λ) = 0 if G0(λ) ≥ (1 − ε)G1(λ). From these relations we can conclude that the following theorems hold true. Theorem 8.1. Let the spectral density matrices F 0(λ) ∈ DU V , G0(λ) ∈ Dε satisfy the minimality condition (1). Let condition (22) be satisfied. These spectral density matrices F 0(λ), G0(λ) are least favorable in the class DU V × Dε for the optimal linear extrapolation of the functional A�ξ if they satisfy conditions (45)–(50) and determine a solution to the extremum problem (35). The spectral characteristic h(F 0, G0) calculated by the formula (24) is minimax-robust for the optimal linear extrapolation of the functional A�ξ. Theorem 8.2. Let the spectral density matrix F (λ) be known and let spec- tral density matrices F (λ), G0(λ) ∈ Dε satisfy the minimality condition (1). Let condition (22) be satisfied. The spectral density matrix G0(λ) is least fa- vorable in the class Dε for the optimal linear extrapolation of the functional A�ξ if G0(λ) = max { (1 − ε)G1(λ), �α−1(A(λ)F (λ) − C0(λ)) − G(λ) } ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 109 and (F (λ), G0(λ)) determine a solution to the extremum problem (35). The spectral characteristic h(F, G0) calculated by the formula (24) is minimax- robust for the optimal linear extrapolation of the functional A�ξ. 9. Least favorable spectral densities in the class D0. Consider the problem of minimax extrapolation of the functional A�ξ based on observations �ξ(t) for t < 0 under the condition that spectral den- sity matrix F (λ) is from the set of spectral density matrices D0 = { F (λ) : 1 2π ∫ ∞ −∞ F (λ)dλ = P } . With the help of the Lagrange multipliers method we can find that solutions to the conditional extremum problem (36) that determine the least favorable density matrix F 0(λ) ∈ D0 of the maximal rank is of the form (51) F 0(λ) = �β [∫ ∞ 0 (Ad)(t)eitλdt ] · [∫ ∞ 0 (Ad)(t)eitλdt ]∗ �β∗. The unknown �β = (β1, . . . , βn)� and {d(t), t ≥ 0} are determined by the canonical factorization (39) of the density matrix F 0(λ) and the condition F 0(λ) ∈ D0. For solutions d = d(t), t ≥ 0 to the system of equations (52) (Ad)(t) = �c d∗(t), t ≥ 0, �c = (c1, . . . , cn), such that (53) 1 2π ∫ ∞ 0 d(t)d∗(t) dt = P, the following equality holds true (54) F (λ) = [∫ ∞ 0 d(t)e−itλdt ] [∫ ∞ 0 d(t)e−itλdt ]∗ = �β [∫ ∞ 0 (Ad)(t)eitλdt ] [∫ ∞ 0 (Ad)(t)eitλdt ]∗ �β∗. Denote by νP 0 the maximal value of ‖Ad‖2, where d = {d(t), t ≥ 0} are solutions to equation (52) that satisfy condition (53) and determine the canonical factorization (39) of the density matrix of the maximal rank F (λ), F (λ) ∈ D0. Denote by μP 0 the maximal value of ‖Ad‖2, where d = {d(t), t ≥ 0} satisfy condition (53) and determine the canonical fac- torization (39) of the density matrix F 0(λ) ∈ D0. If there exists a solution d0 = {d0(t), t ≥ 0} to the equation (52) that satisfy condition (53) and such that νP 0 = μP 0 = ‖Ad0‖2 , then the least favorable in D0 is the density matrix of the maximal rank (39). The stationary process �ξ(t) in this case admits the moving average representation (28). The minimax (robust) spec- tral characteristics of the optimal linear extrapolationof the functional A�ξ 110 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA is calculated by formula (31) since the functions A(λ) and r(λ) are bounded and h(F 0) ∈ HD. The following theorem holds true. Theorem 9.1. If there exists a solution d0 = {d0(t), t ≥ 0} to equation (52) that satisfy condition (53) and such that νP 0 = μP 0 = ‖Ad0‖2 , then the least favorable in D0 for the optimal linear extrapolation of the functional A�ξ is the density matrix of the maximal rank (39). If νP 0 < μP 0 , then the least favorable in D0 density matrix is determined by conditions (39), (40). The corresponding stationary process �ξ(t) in this case admits the moving average representation (28). The minimax spectral characteristics h(F ) of the optimal linear extrapolation is calculated by formula (31). For the functional AL �ξ the density matrix (51) is of the form (55) F 0(λ) = �β ⎡ ⎣ L∫ 0 (ALd)(t)eitλdt ⎤ ⎦ ⎡ ⎣ L∫ 0 (ALd)(t)eitλdt ⎤ ⎦ ∗ �β∗. In this case the equality holds true rL(λ)r∗L(λ) = [ L∫ 0 (ALd)(t)eitλdt ] [ L∫ 0 (ALd)(t)eitλdt ]∗ = = [ L∫ 0 (ÃLd)(t)eitλdt ] [ L∫ 0 (ÃLd)(t)eitλdt ]∗ , where (ÃLd)(t) = t∫ 0 �a(L − t + u)d(u)du. For these reasons for all solutions d = {d(t), 0 ≤ t ≤ L} to equations (56) (ALd)(t) = �cd∗(t), 0 ≤ t ≤ L, �c = (c1, . . . , cT ), (57) (ÃLd)(t) = �bd∗(t), 0 ≤ t ≤ L, �b = (b1, . . . , bT ), such that (58) L∫ 0 d(t)d∗(t)dt = P, the equality holds true F (λ) = ⎡ ⎣ L∫ 0 d(t)e−itλdt ⎤ ⎦ ⎡ ⎣ L∫ 0 d(t)e−itλdt ⎤ ⎦ ∗ = �β rL(λ)r∗L(λ)�β∗. Denote by νLP 0 the maximal value of ‖ALd‖2 = ∥∥∥ÃLd ∥∥∥2 , where d = {d(t), 0 ≤ t ≤ L} are solutions to equations (56), (57) that satisfy condition (58) and determine the canonical factorization (29) of the density matrix F 0(λ). ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 111 Denote by μLP 0 the maximal value of ‖ALd‖2, where d = {d(t), 0 ≤ t ≤ L} satisfy condition (58) and determine the canonical factorization (29) of the density matrix F 0(λ) with F 0(λ) of the form (55). If there exists a solution d0 = {d0(t), 0 ≤ t ≤ L} to equation (56), or equation (57), that satisfy condition (58) and such that νLP 0 = μLP 0 = ‖ALd0‖2 , then the least favorable in D0 is the density matrix (59) F 0(λ) = ⎡ ⎣ L∫ 0 d0(t)e−itλdt ⎤ ⎦ ⎡ ⎣ L∫ 0 d0(t)e−itλdt ⎤ ⎦ ∗ . The following theorem holds true. Theorem 9.2. If there exists a function d0 = {d0(t), 0 ≤ t ≤ L}, that satisfy condition (58) and such that νLP 0 = μLP 0 = ‖ALd0‖2 , then the least favorable in D0 for the optimal linear estimation of the functional AL �ξ is the density matrix (59). The corresponding stationary process �ξ(t) in this case admits the moving average representation (44). If νLP 0 < μLP 0 , then the density matrix (55), that admits the canonical factorization (42), is the least favorable in D0. The vector �β and the function d0 = {d0(t), 0 ≤ t ≤ L} are determined by conditions (43), (58). The minimax spectral characteristics h(F ) of the optimal linear estimate of the functional AL �ξ is calculated by formula (33). Example 4. Consider the problem for the functional A1 �ξ = 1∫ 0 �a(t)�ξ(t)dt, where �ξ is a two-dimensional stochastic process. The least favorable in D0 for the optimal linear estimation of the functional A1 �ξ is the density matrix F (λ) = ⎡ ⎣ 1∫ 0 d(t)e−itλdt ⎤ ⎦ · ⎡ ⎣ 1∫ 0 d(t)e−itλdt ⎤ ⎦ ∗ , where the matrix function d = {dkj(t), 0 ≤ t ≤ 1; k, j = 1, 2} is a solution to the conditional extremum problem 1∫ 0 1∫ 0 min(x,y)∫ 0 �a(y)d(y − u)d∗(x − u)�a∗(x)dydx → max, 1∫ 0 d(t)d∗(t)dt = P. 112 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA The corresponding process �ξ(t) in this case admits the canonical moving average representation of the form �ξ(t) = t∫ t−1 d(t − u)d�ε(u). For more results on minimax-robust extrapolation of multidimensional stationary processes in the case of observations without noise see article [16]. For the corresponding results for multidimensional stationary sequences see article [15]. 10. Conclusions We propose formulas for calculation the mean square errors and the spec- tral characteristic of the optimal linear estimate of the unknown value of the functional AL �ξ = ∫ L 0 �a(t)�ξ(t)dt which depends on the unknown values of a multidimensional stationary stochastic process �ξ(t) based on observations of the process �ξ(t) + �η(t) for t ∈ R\[0, L], where �η(t) is uncorrelated with �ξ(t) multidimensional stationary process, and formulas for calculation the mean square error and the spectral characteristic of the optimal linear extrapo- lation of the unknown value of the functional A�ξ = ∫∞ 0 �a(t)�ξ(t)dt which depends on the unknown values of a multidimensional stationary stochas- tic process �ξ(t) based on observations of the process �ξ(t) + �η(t) for t < 0 under the condition that spectral density matrices F (λ) and G(λ) of the signal process �ξ(t) and the noise process �η(t) are known exactly. Formu- las are proposed that determine the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of the functionals for some classes D = DF ×DG of spectral density matrices un- der the condition that spectral density matrices F (λ), G(λ) are not known, but classes D = DF × DG of admissible spectral matrices are given. References 1. Franke, J., On the robust prediction and interpolation of time series in the pres- ence of correlated noise, J. Time Series Analysis, 5 (1984), no. 4, 227–244. 2. Franke, J., Minimax robust prediction of discrete time series, Z. Wahrsch. Verw. Gebiete. 68 (1985), 337–364. 3. Franke, J., Poor, H. V. Minimax–robust filtering and finite–length robust predic- tors, In Robust and Nonlinear Time Series Analysis (Heidelberg, 1983) Lecture Notes in Statistics, Springer-Verlag, 26 (1984), 87–126. 4. Franke, J., A general version of Breiman’s minimax filter, Note di Matematica, 11 (1991), 157–175. 5. Grenander, U., A prediction problem in game theory, Ark. Mat., 3 (1957), 371– 379. 6. Kailath, T., A view of three decades of linear filtering theory, IEEE Trans. on Inform. Theory, 20 (1974), no. 2, 146–181. 7. Kassam, S. A., Poor, H. V. Robust techniques for signal processing: A survey, Proc. IEEE, 73 (1985), no. 3, 433–481. ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 113 8. Kolmogorov, A. N., Selected works of A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics., Ed. by A. N. Shiryayev. Mathematics and Its Applications. Soviet Series. 26. Dordrecht etc.: Kluwer Academic Publishers, (1992). 9. Moklyachuk, M. P., Stochastic autoregressive sequence and minimax interpola- tion, Theor. Probab. and Math. Stat., 48, (1994), 95–103. 10. Moklyachuk, M. P., Estimates of stochastic processes from observations with noise, Theory Stoch. Process., 3(19) (1997), no.3-4, 330–338. 11. Moklyachuk, M. 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A., Stationary stochastic processes, 2nd rev. ed. Moscow, “Nauka”, 1990. (English transl. of 1st ed., Holden-Day, San Francisco, 1967) 19. Vastola, K. S., Poor, H. V., An analysis of the effects of spectral uncertainty on Wiener filtering, Automatica, 28 (1983), 289–293. 20. Wiener, N., Extrapolation, interpolation, and smoothing of stationary time se- ries. With engineering applications, Cambridge, Mass.: The M. I. T. Press, Massachusetts Institute of Technology. (1966). 21. Yaglom, A. M., Correlation theory of stationary and related random functions. Vol. I: Basic results. Springer Series in Statistics. New York etc.: Springer-Verlag. (1987). 22. Yaglom, A. M., Correlation theory of stationary and related random functions. Vol. II: Supplementary notes and references. Springer Series in Statistics. New York etc.: Springer-Verlag. (1987). Department of Probability Theory and Mathematical Statistics, Kyiv National Taras Shevchenko University, Kyiv 01033, Ukraine E-mail address: mmp@univ.kiev.ua

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