Minimax-robust filtering problem for stochastic ... - Semantic Scholar

STATISTICS,OPTIMIZATION AND INFORMATION COMPUTING Stat., Optim. Inf. Comput., Vol. 2, September 2014, pp 176–199. Published online in International Academic Press (www.IAPress.org)

Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences Maksym Luz 1 and Mikhail Moklyachuk 1, ∗ 1 Department

of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Ukraine.

Received 25 December 2013; Accepted 24 April 2014 Editor: Antanas Zilinskas

The problem of optimal estimation of the linear functionals Aξ = ∑ a(k)ξ(−k) and AN ξ = N k=0 a(k)ξ(−k) which depend on the unknown values k=0 of a stochastic sequence ξ(k) with stationary nth increments is considered. Estimates are based on observations of the sequence ξ(k) + η(k) at points of time k = 0, −1, −2, . . ., where the sequence η(k) is stationary and uncorrelated with the sequence ξ(k). Formulas Abstract ∑∞

for calculating the mean-square errors and spectral characteristics of the optimal estimates of the functionals are proposed in the case of spectral certainty, where spectral densities of sequences ξ(k) and η(k) are exactly known. Minimax (robust) method of estimation is applied in the case where spectral densities of the sequences are not known exactly, but sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special classes of admissible densities. The filtering problem for a class of cointegrated sequences is investigated. Keywords Stochastic sequence with stationary increments; cointegrated sequence; minimax-robust estimate; mean square error; least favorable spectral density; minimax-robust spectral characteristic

DOI: 10.19139/soic.v2i3.56

∗ Correspondence to: Department of Probability Theory, Statistics and Actuarial Mathematics, Taras

Shevchenko National University of Kyiv, Kyiv 01601, Ukraine. E-mail: [email protected]

ISSN 2310-5070 (online) ISSN 2311-004X (print) c 2014 International Academic Press Copyright ⃝

MINIMAX-ROBUST FILTERING PROBLEM FOR STOCHASTIC SEQUENCES

177

1. Introduction Wide sense stationary and related stochastic processes have been studied by many scientists. The obtained results found their applications for solving actual problems in description and analysis of models of economic and financial time series. The most simple examples are linear stationary models such as moving average (MA), autoregressive (AR) or autoregressive-moving average (ARMA) sequences, state space model, all of which refer to stationary sequences with rational spectral function without unit AR-roots. Time series with trends and seasonal components are modeled by integrated ARMA (ARIMA) sequences which have unit roots in their autoregressive parts and are examples of sequences with stationary increments. Such models attract interest of scientists during the last 30 years. The main points concerning model definition, parameter estimation, forecasting and further investigation are discussed in the well-known book by Box, Jenkins and Reinsel [1]. While analyzing financial data some economists noticed that in special cases linear combinations of integrated sequences become stationary. Grander [10] called this phenomenon cointegration. Cointegrated sequences found their application in applied and theoretical econometrics and financial time series analysis [9]. Estimation of unknown values of stochastic processes is an important part of the theory of stochastic processes. Effective methods of solution of the linear extrapolation, interpolation and filtering problems for stationary stochastic processes were developed by Kolmogorov [15], Wiener [37], Yaglom [38, 39]. Further results one can find in book by Rozanov [35]. Yaglom [40, 41] developed theory of non-stationary processes whose increments of order n form a stationary process. Further results for such stochastic processes were presented by Pinsker [33], Yaglom and Pinsker [32]. See books by Yaglom [38, 39] for more relative results and references. The mean square optimal estimation problems for stochastic processes with nth stationary increments are natural generalizations of extrapolation, interpolation and filtering problems for stationary stochastic processes. The classical methods of solution of extrapolation, interpolation and filtering problems are based on the assumption that spectral density of the process is known. In practice, however, it is impossible to obtain complete information on the spectral density in most cases. To solve the problem one finds parametric or nonparametric estimates of the unknown spectral density or selects a density by other reasoning. Then the classical estimation method is applied provided that the estimated or selected density is the true one. Vastola and Poor [36] have demonstrated that the described procedure can result in significant increasing of the value of error. This is a reason for searching estimates which are optimal for all densities from a certain class of admissible spectral densities. These estimates are called minimax since they minimize the maximal value of the error of estimates. A survey of results Stat., Optim. Inf. Comput. Vol. 2, September 2014.

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MAKSYM LUZ AND MIKHAIL MOKLYACHUK

in minimax (robust) methods of data processing can be found in the paper by Kassam and Poor [14]. The paper by Grenander [11] should be marked as the first one where the minimax extrapolation problem for stationary processes was formulated and solved. Franke and Poor [13], Franke [12] investigated the minimax extrapolation and filtering problems for stationary sequences with the help of convex optimization methods. In papers by Moklyachuk [20] – [22] the minimax approach was applied to extrapolation, interpolation and filtering problems for functionals which depend on the unknown values of stationary processes and sequences. For more results and details see, for example, book by Moklyachuk [29], articles and book by Moklyachuk and Masyutka [26] − [31]. Dubovets’ka and Moklyachuk [3] − [8] investigated the minimax-robust estimation problems (extrapolation, interpolation and filtering) for periodically correlated stochastic processes. In papers [16] − [19], [25] by Luz and Moklyachuk the minimax interpolation and extrapolation problems for linear functionals which depend on unknown (missed) values of stochastic process ξ(m) with stationary nth increments from observations of the process with and without noise were investigated. In the paper [18] they ∑ investigated the problem of optimal linear estimation of ∞ the functional Aξ = k=0 a(k)ξ(−k) which depends on unknown values of a stochastic sequence ξ(k) with nth stationary increments from observations of the sequence ξ(k) + η(k) at points k = 0, −1, −2, . . ., where η(k) is a stochastic sequence with stationary nth increments which is uncorrelated with the sequence ξ(k). To solve the problem the functional Aξ is represented as a functional of increments of the sequence providing some conditions on the coefficients a(k), k ≥ 0. This representation gives a possibility to find an estimate only ∑N for the functional Aξ , but not for the functional AN ξ = k=0 a(k)ξ(−k). In the present article we require the noise sequence η(k) to be stationary. This ∑∞ condition let us find estimates of both functionals Aη = k=0 a(k)η(−k) and ∑N AN η = k=0 a(k)η(−k) which are used for solving the filtering problem for functionals Aξ and AN ξ . The obtained results give us a method of solution the filtering problem for cointegrated sequences ξ(k) and ζ(k) under the condition that a stationary linear combination of the sequences does not correlate with the sequence ξ(k). The estimation problem is solved in the case of spectral certainty where spectral densities of sequences ξ(k) and η(k) are exactly known as well as in the case of spectral uncertainty where spectral densities of the sequences are not exactly known, but a set of admissible spectral densities is given. Formulas that determine least favorable spectral densities and minimax (robust) spectral characteristics of optimal estimates of functionals are proposed in the case of spectral uncertainty for concrete classes of admissible spectral densities.

Stat., Optim. Inf. Comput. Vol. 2, September 2014.


179

2. Stationary increment stochastic sequences. Spectral representation Definition 2.1 For a given stochastic sequence {ξ(m), m ∈ Z} the sequence ξ (n) (m, µ) = (1 − Bµ )n ξ(m) =

n ∑ (−1)l Cnl ξ(m − lµ),

(1)

l=0

where Bµ is a backward shift operator with step µ ∈ Z, such that Bµ ξ(m) = ξ(m − µ), is called stochastic nth increment sequence with step µ ∈ Z. For the stochastic nth increment sequence ξ (n) (m, µ) the following relations hold true ξ (n) (m, −µ) = (−1)n ξ (n) (m + nµ, µ), (2) ∑(k−1)n Al ξ (n) (m − lµ, µ), k ∈ N, (3) ξ (n) (m, kµ) = l=0

where coefficients {Al , l = 0, 1, 2, . . . , (k − 1)n} are determined by the representation (k−1)n ∑ (1 + x + . . . + xk−1 )n = Al xl . l=0

Definition 2.2 The stochastic nth increment sequence ξ (n) (m, µ) generated by stochastic sequence {ξ(m), m ∈ Z} is wide sense stationary if the mathematical expectations Eξ (n) (m0 , µ) = c(n) (µ), Eξ (n) (m0 + m, µ1 )ξ (n) (m0 , µ2 ) = D(n) (m, µ1 , µ2 )

exist for all m0 , µ, m, µ1 , µ2 and do not depend on m0 . The function c(n) (µ) is called mean value of the nth increment sequence and the function D(n) (m, µ1 , µ2 ) is called structural function of the stationary nth increment sequence (or structural function of nth order of the stochastic sequence {ξ(m), m ∈ Z}). The stochastic sequence {ξ(m), m ∈ Z} which determines the stationary nth increment sequence ξ (n) (m, µ) by formula (1) is called sequence with stationary nth increments (or integrated sequence of order n). Theorem 2.1 The mean value c(n) (µ) and the structural function D(n) (m, µ1 , µ2 ) of the stochastic stationary nth increment sequence ξ (n) (m, µ) can be represented in the following forms c(n) (µ) = cµn , (4) ∫ π 1 D(n) (m, µ1 , µ2 ) = eiλm (1 − e−iµ1 λ )n (1 − eiµ2 λ )n 2n dF (λ), (5) λ −π Stat., Optim. Inf. Comput. Vol. 2, September 2014.

180


where c is a constant, F (λ) is a left-continuous nondecreasing bounded function with F (−π) = 0. The constant c and the function F (λ) are determined uniquely by the increment sequence ξ (n) (m, µ). From the other hand, a function c(n) (µ) which has form (4) with a constant c and a function D(n) (m, µ1 , µ2 ) which has form (5) with a function F (λ) which satisfies the indicated conditions are the mean value and the structural function of a stationary nth increment sequence ξ (n) (m, µ). Using representation (5) of the structural function of a stationary nth increment sequence ξ (n) (m, µ) and the Karhunen theorem [2], we obtain the following spectral representation of the stationary nth increment sequence ξ (n) (m, µ): ∫ π 1 ξ (n) (m, µ) = dZ(λ), (6) eimλ (1 − e−iµλ )n n (iλ) −π where Z(λ) is an orthogonal stochastic measure on [−π, π) connected with the spectral function F (λ) by the relation EZ(A1 )Z(A2 ) = F (A1 ∩ A2 ) < ∞.

(7)

3. Filtering problem Let a stochastic sequence {ξ(m), m ∈ Z} define stationary nth increment sequence ξ (n) (m, µ) with absolutely continuous spectral function F (λ) which has spectral density f (λ). Let {η(m), m ∈ Z} be an uncorrelated with the sequence ξ(m) stationary stochastic sequence with absolutely continuous spectral function G(λ) which has spectral density g(λ). Without loss of generality we will assume that mean values of the increment sequence ξ (n) (m, µ) and stationary sequence η(m) equal to 0. Consider the problem of mean-square optimal linear estimation of the functionals ∞ N ∑ ∑ Aξ = a(k)ξ(−k), AN ξ = a(k)ξ(−k) k=0

k=0

which depend on unknown values of the sequence ξ(m) from observations of the sequence ζ(m) = ξ(m) + η(m) at points m = 0, −1, −2, . . .. We will consider the case where the step µ > 0. And we will suppose that conditions ∞ ∑ k=0

|aµ (k − µn)| < ∞,

∞ ∑ (k + 1)|aµ (k − µn)|2 < ∞

(8)

k=0

hold true for coefficients aµ (k), k ≥ −µn, which are defined in the following part of the paper. Stat., Optim. Inf. Comput. Vol. 2, September 2014.


181

The functional Aξ can be represented in the form Aξ = Aζ − Aη,

∑∞

∑∞ where Aζ = k=0 a(k)ζ(−k), Aη = k=0 a(k)η(−k). b = E|Aξ − Aξ| b 2 denote the mean-square error of the estimate Let ∆(f, g, Aξ) b of the functional Aξ and let ∆(f, g, Aη) b = E|Aη − Aη| b 2 denote the meanAξ b of the functional Aη . Since the functional Aζ is square error of the estimate Aη determined by the observed values of the sequence ζ(m), the following relations hold true b = Aζ − Aη, b Aξ (9) b = E|Aξ − Aξ| b 2 = E|Aζ − Aη − Aζ + Aη| b 2 = E|Aη − Aη| b 2 = ∆(f, g, Aη). b ∆(f, g, Aξ)

To find the mean-square optimal estimate of the functional Aη we use spectral representations of the stationary sequence η(m), the stationary nth increment sequence η (n) (m, µ) and apply the Hilbert space orthogonal projection method proposed by Kolmogorov [15]. The stationary stochastic sequence η(m) admits the spectral representation ∫ π eiλm dZη (λ) η(m) = −π

and the nth increment sequence η (n) (m, µ) admits the spectral representation ∫ π (n) eiλm (1 − e−iλµ )n dZη (λ), η (m, µ) = −π

where Zη (λ) is an orthogonal stochastic measure on [−π, π) corresponding to the spectral function G(λ). The stationary increment sequence ζ (n) (m, µ) admits the spectral representation ∫ π 1 eiλm (1 − e−iλµ )n ζ (n) (m, µ) = dZξ(n) +η(n) (λ) n (iλ) −π ∫

π

=

e −π

iλm

(1 − e

1 ) dZ (n) (λ) + (iλ)n ξ

−iλµ n

∫

π

−π

eiλm (1 − e−iλµ )n dZη (λ),

where dZη(n) (λ) = (iλ)n dZη (λ), λ ∈ [−π, π). The spectral density p(λ) of the sequence ζ(m) is determined by spectral densities f (λ) and g(λ) by the relation p(λ) = f (λ) + λ2n g(λ). Stat., Optim. Inf. Comput. Vol. 2, September 2014.

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MAKSYM LUZ AND MIKHAIL MOKLYACHUK (n)

(n)

Denote by H 0 (ξµ + ηµ ) the closed linear subspace of the Hilbert space H = L2 (Ω, F, P) of random variables of the second order which is generated by values {ξ (n) (k, µ) + η (n) (k, µ) : k ≤ 0}, µ > 0. Denote by L02 (f (λ) + λ2n g(λ)) the closed linear subspace of the Hilbert space L2 (f (λ) + λ2n g(λ)) generated by functions {eiλk (1 − e−iλµ )n

It follows from the formula

∫

π

ξ (n) (k, µ) + η (n) (k, µ) = −π

1 : k ≤ 0}. (iλ)n

eiλk (1 − e−iλµ )n

1 dZ (n) (n) (λ) (iλ)n ξ +η

that there exists a one to one correspondence between elements eiλk (1 − 1 e−iλµ )n of the space L02 (f (λ) + λ2n g(λ)) and elements ξ (n) (k, µ) + (iλ)n (n) (n) η (n) (k, µ) of the space H 0 (ξµ + ηµ ). b of the functional Aξ admits the representation Every linear estimate Aξ ∫ π b = Aζ − hµ (λ)dZξ(n) +η(n) (λ), (10) Aξ −π

b . The mean square where hµ (λ) is the spectral characteristic of the estimate Aη b is a projection of the element Aη on the subspace H 0 (ξµ(n) + optimal estimate Aη (n) ηµ ). This estimate is determined by the following conditions: b ∈ H 0 (ξµ(n) + ηµ(n) ); 1) Aη b ⊥ H 0 (ξµ(n) + ηµ(n) ). 2) (Aη − Aη) It comes from condition 2) that for all k ≤ 0 the following relations hold true (n) (k, µ) + η (n) (k, µ)) b E(Aη − Aη)(ξ

1 = 2π

∫

−π

1 − 2π

(

π

∫

) A(e−iλ ) − (iλ)n hµ (λ) e−iλk (1 − eiλµ )n g(λ)dλ

π

−π

hµ (λ)e−iλk (1 − eiλµ )n

1 f (λ)dλ = 0. (−iλ)n

These relations can be represented in the form ∫ π [ ] (1 − eiλµ )n −iλk A(e−iλ )g(λ)(−iλ)n − hµ (λ)(f (λ) + λ2n g(λ)) e dλ (−iλ)n −π = 0, k ≤ 0. Stat., Optim. Inf. Comput. Vol. 2, September 2014.


183

From these equations we can derive the spectral characteristic hµ (λ) of the b . It is of the form estimate Aη hµ (λ) = A(e−iλ ) A(e−iλ ) =

(−iλ)n Cµ (eiλ ) (−iλ)n g(λ) − , f (λ) + λ2n g(λ) (1 − eiλµ )n (f (λ) + λ2n g(λ)) ∞ ∑

a(k)e−iλk ,

Cµ (eiλ ) =

k=0

∞ ∑

cµ (k)eiλk .

k=1

where cµ (k) are unknown coefficients to be determined. It follows from b admits the condition 1) that the spectral characteristic hµ (λ) of the estimate Aη representation hµ (λ) = h(λ)(1 − e−iλµ )n

where

∫

π

−π

∫

π −π

[

1 , (iλ)n

|h(λ)|2 |1 − eiλµ |2n

h(λ) =

∞ ∑

s(k)e−iλk ,

k=0

f (λ) + λ2n g(λ) dλ < ∞, λ2n

(iλ)n hµ (λ) ∈ L02 , (1 − e−iλµ )n

]

λ2n Cµ (eiλ ) A(e−iλ )λ2n g(λ) − e−iλl dλ = 0, l ≥ 1. −iλµ n 2n iλµ (1 − e ) (f (λ) + λ g(λ)) |1 − e |2n (f (λ) + λ2n g(λ))

(11)

Suppose that spectral densities f (λ), g(λ) are such that the following condition holds true: ∫ π λ2n dλ < ∞. (12) iλµ 2n | (f (λ) + λ2n g(λ)) −π |1 − e Determine for every k, j ∈ Z the Fourier coefficients of the corresponding functions ∫ π λ2n g(λ) 1 µ e−iλ(j+k) Sk,j = dλ; iλµ 2n 2π −π |1 − e | (f (λ) + λ2n g(λ)) ∫ π λ2n 1 µ eiλ(j−k) Pk,j = dλ; 2π −π |1 − eiλµ |2n (f (λ) + λ2n g(λ)) ∫ π f (λ)g(λ) 1 eiλ(j−k) dλ. Qk,j = 2π −π (f (λ) + λ2n g(λ)) Using these Fourier coefficients we can represent equation (11) in the form of the following system of linear equations ∞ ∑ m=−µn

µ Sl,m aµ (m) =

∞ ∑

µ Pl,k cµ (k),

l ≥ 1,

k=1


184


where aµ (m) =

n ∑ l=max{[

(−1)l Cnl a(m + µl),

m ≥ −µn,

(13)

′ −m ,0 µ

] }

Here [x]′ denotes the least integer number among numbers which are greater or equal to x. This system can be written in the form Sµ aµ = Pµ cµ ,

where cµ = (cµ (1), cµ (2), cµ (2), . . .)′ , aµ = (aµ (−µn), aµ (−(µn − 1)), . . .)′ , Pµ , Sµ are linear operators in the space ℓ2 defined by the matrices with elements µ µ (Pµ )l,k = Pl,k , l, k ≥ 1, (Sµ )l,m = Sl,m , l ≥ 1, m ≥ −µn. Consequently, the unknown coefficients cµ (k) which determine the spectral characteristic hµ (λ) are calculated by the formula cµ (k) = (P−1 µ Sµ aµ )k ,

k ≥ 0,

−1 where (P−1 µ Sµ aµ )k , k ≥ 0, is the k th element of the vector Pµ Sµ aµ . Thus, the b of the functional Aη is spectral characteristic hµ (λ) of the optimal estimate Aη calculated by the formula ∑∞ iλk (−iλ)n k=1 (P−1 (−iλ)n g(λ) µ Sµ aµ )k e −iλ hµ (λ) = A(e ) − . (14) f (λ) + λ2n g(λ) (1 − eiλµ )n (f (λ) + λ2n g(λ))

The mean-square error of the estimate is calculated by the formula b = ∆(f, g; Aη) b = E|Aη − Aη| b 2 ∆(f, g; Aξ) ∑∞ ∫ π iλk 2 A(e−iλ )(1 − eiλµ )n f (λ) + λ2n k=1 (P−1 1 µ Sµ aµ )k e = g(λ)dλ 2π −π |1 − eiλµ |2n (f (λ) + λ2n g(λ))2 ∑∞ ∫ π iλk 2 A(e−iλ )(1 − eiλµ )n λ2n g(λ) − λ2n k=1 (P−1 1 µ S µ aµ ) k e + f (λ)dλ 2π −π λ2n |1 − eiλµ |2n (f (λ) + λ2n g(λ))2 = ⟨Sµ aµ , P−1 µ Sµ aµ ⟩ + ⟨Qa, a⟩,

(15)

where a = (a(0), a(1), a(2), . . .)′ , Q is a linear operator in the space ℓ2 defined by the matrix with elements (Q)l,k = Ql,k , l, k ≥ 0. These observations can be summarized in the form of the theorem. Theorem 3.1 Let {ξ(m), m ∈ Z} be a stochastic sequence which defines stationary nth increment sequence ξ (n) (m, µ) with absolutely continuous spectral function F (λ) which has spectral density f (λ). Let {η(m), m ∈ Z} be an uncorrelated with the sequence ξ(m) stationary stochastic sequence with absolutely continuous spectral Stat., Optim. Inf. Comput. Vol. 2, September 2014.

185


function G(λ) which has spectral density g(λ). Let condition (12) be satisfied. Let coefficients {aµ (k) : k ≥ −µn} defined by formula (13) satisfy conditions b of the functional Aξ of unknown elements (8). The optimal linear estimate Aξ ξ(m), m ≤ 0, based on observations of the sequence ξ(m) + η(m) at points m = 0, −1, −2, . . . is calculated by formula (10). The spectral characteristic hµ (λ) b is calculated by formula (14). The value of the meanof the optimal estimate Aξ b square error ∆(f, g; Aξ) is calculated by formula (15). bN ξ of the functional We can use Theorem 3.1 to obtain the optimal estimate A AN ξ of unknown elements ξ(m), m = 0, −1, −2, . . . , −N , from observations of the sequence ξ(m) + η(m) at points m = 0, −1, −2, . . .. Take a(k) = 0, k > N . Then the spectral characteristic hµ,N (λ) of the linear estimate ∫ π b AN ξ = AN ζ − hµ,N (λ)dZξ(n) +η(n) (λ) (16) −π

is calculated by the formula hµ,N (λ) = AN (e−iλ )

∑∞ iλk (−iλ)n k=1 (P−1 (−iλ)n g(λ) µ Sµ,N aµ,N )k e − , 2n iλµ n 2n f (λ) + λ g(λ) (1 − e ) (f (λ) + λ g(λ)) (17)

where AN (e−iλ ) =

N ∑

a(k)e−iλk ,

k=0

aµ,N = (aµ,N (−µn), aµ,N (−(µn − 1)), . . . , aµ,N (N ), 0, . . .), min{[ N −m ],n} µ

aµ,N (m) =

∑

(−1)l Cnl a(m + µl),

−µn ≤ m ≤ N,

(18)

′ −m ,0 µ

l=max{[

] }

Sµ,N is a linear operator in the space ℓ2 defined by the matrix with elements µ (Sµ,N )l,m = Sl,m , l ≥ 1, −µn ≤ m ≤ N , and (Sµ,N )l,m = 0, l ≥ 1, m > N . The bN ξ is calculated by the formula mean-square error of the estimate A bN ξ) = ∆(f, g; A bN η) = E|AN η − A bN η|2 ∆(f, g; A =

+

1 2π

1 2π

π

iλk 2 AN (e−iλ )(1 − eiλµ )n f (λ) + λ2n ∑∞ (P−1 µ Sµ,N aµ,N )k e k=1

−π

|1 − eiλµ |2n (f (λ) + λ2n g(λ))2

∫

g(λ)dλ

π

∑ iλk 2 AN (e−iλ )(1 − eiλµ )n λ2n g(λ) − λ2n ∞ (P−1 µ Sµ,N aµ,N )k e k=1

−π

λ2n |1 − eiλµ |2n (f (λ) + λ2n g(λ))2

∫

= ⟨Sµ,N aµ,N , P−1 µ Sµ,N aµ,N ⟩ + ⟨QN aN , aN ⟩,

f (λ)dλ

(19)


186


where aN = (a(0), a(1), . . . , a(N ), 0, . . .), QN is a linear operator in the space ℓ2 defined by the matrix with elements (QN )l,k = Qµl,k , 0 ≤ l, k ≤ N , and (QN )l,k = 0 otherwise. The following theorem holds true. Theorem 3.2 Let {ξ(m), m ∈ Z} be a stochastic sequence which defines stationary nth increment sequence ξ (n) (m, µ) with an absolutely continuous spectral function F (λ) which has spectral density f (λ). Let {η(m), m ∈ Z} be an uncorrelated with the sequence ξ(m) stationary stochastic sequence with an absolutely continuous spectral function G(λ) which has spectral density g(λ). Let condition (12) be satisfied. Let coefficients {aµ,N (k) : −µn ≤ k ≤ N } be defined by formula (18). bN ξ of the functional AN ξ of unknown elements The optimal linear estimate A ξ(k), k = 0, −1, −2, . . . , −N , from observations of the sequence ξ(m) + η(m) at points m = 0, −1, −2, . . . is calculated by formula (16). The spectral characteristic bN ξ is calculated by formula (17). The value of hµ,N (λ) of the optimal estimate A bN ξ) is calculated by formula (19). the mean-square error ∆(f, g; A A particular case of the considered problem is the problem of estimation of an unobserved value ξ(−p) at a point −p, p ≥ 0, from observations of the sequence ξ(k) + η(k) at points k = 0, −1, −2, . . .. In this case the vector aµ,N has coefficients aµ,N (m) = (−1)l Cnl if m = p − µl, l = 0, 1, 2, . . . , n, m = −µn, −(µn − 1), . . ., and aµ,N (m) = 0 if m ̸= p − µl, l = 0, 1, 2, . . . , n, m = −µn, −(µn − 1), . . .. Let us define a vector an = (an (0), an (1), . . . , an (n), 0, 0, . . .)′ , where an (k) = (−1)k Cnk , k = 0, 1, 2, . . . , n. It follows from the derived formulas that the spectral characteristic hµ,p (λ) of the optimal estimate ∫ π b hµ,p (λ)dZξ(n) +η(n) (λ) (20) ξ(−p) = ζ(−p) − −π

of the value ξ(−p), p ≥ 0, can be calculated by the formula ∑∞ iλk (−iλ)n k=1 (P−1 (−iλ)n g(λ) µ Sµ,p an )k e hµ,p (λ) = e−iλp − , 2n iλµ n 2n f (λ) + λ g(λ) (1 − e ) (f (λ) + λ g(λ))

(21)

where Sµ,p is a linear operator in the space ℓ2 defined by the matrix with µ , l ≥ 1, 0 ≤ k ≤ n, and (Sµ,N )l,k = 0, l ≥ 1, k > n. elements (Sµ,p )l,k = Sl,p−µk The mean-square error of the estimate is calculated by the formula b ∆(f, g; ξ(−p)) = ∆(f, g; ηb(−p)) = E|η(−p) − ηb(−p)|2 1 = 2π

∫

π

−π

−iλp ∑∞ iλk 2 e (1 − eiλµ )n f (λ) + λ2n k=1 (P−1 µ Sµ,p an )k e g(λ)dλ |1 − eiλµ |2n (f (λ) + λ2n g(λ))2 Stat., Optim. Inf. Comput. Vol. 2, September 2014.


1 + 2π

∫

π

−π

187

−iλp ∑∞ iλk 2 e (1 − eiλµ )n λ2n g(λ) − λ2n k=1 (P−1 µ Sµ,p an )k e f (λ)dλ λ2n |1 − eiλµ |2n (f (λ) + λ2n g(λ))2 = ⟨Sµ,p an , P−1 µ Sµ,p an ⟩ + Q0,0 .

(22)

Thus, we have the following statement. Corollary 3.1 b The optimal linear estimate ξ(−p) of the unknown value ξ(−p), p ≥ 0, of a stochastic sequence with nth stationary increments from observations of the sequence ξ(k) + η(k) at points k = 0, −1, −2, . . . is calculated by formula (20). b The spectral characteristic hµ,p (λ) of the optimal estimate ξ(−p) is calculated by b formula (21). The value of the mean-square error ∆(f, g; ξ(−p)) is calculated by formula (22). Theorem 3.1, 3.1 and Corollary 3.1 determine solutions of the filtering problem for the linear functionals Aξ , AN ξ and the value ξ(−p), p ≥ 0, using the Fourier coefficients of functions λ2n , |1 − eiλµ |2n (f (λ) + λ2n g(λ))

λ2n g(λ) . |1 − eiλµ |2n (f (λ) + λ2n g(λ))

However, the problem of finding the inverse operator (Pµ )−1 to the operator Pµ λ2n defined by the Fourier coefficients of the function |1 − eiλµ |2n (f (λ) + λ2n g(λ)) is complicated in most cases. Therefore, we propose a method of finding the operator (Pµ )−1 under the condition that the functions |1 − eiλµ |2n (f (λ) + λ2n g(λ)) , λ2n

λ2n |1 −

eiλµ |2n (f (λ)

+ λ2n g(λ))

(23)

admit the canonical factorizations ∞ 2 |1 − eiλµ |2n (f (λ) + λ2n g(λ)) ∑ −iλk = φ (k)e , µ λ2n

(24)

k=0

∞ 2 ∑ λ2n −iλk = ψ (k)e . µ |1 − eiλµ |2n (f (λ) + λ2n g(λ))

(25)

k=0

Using coefficients φµ (k), ψµ (k), k ≥ 0, from factorizations (24), (25), we define linear operators Φµ and Ψµ in the space ℓ2 . Let (Φµ )k,j = φµ (k − j) and (Ψµ )k,j = ψµ (k − j) for 1 ≤ j ≤ k , (Φµ )k,j = 0 and (Ψµ )k,j = 0 for j > k , k, j ≥ 1. The defined operators admit the following relation: Ψµ Φµ = Φµ Ψµ = I , where I is the identity operator. Moreover, the operator Pµ admits the Stat., Optim. Inf. Comput. Vol. 2, September 2014.

188

MAKSYM LUZ AND MIKHAIL MOKLYACHUK ′

′

factorization Pµ = Ψµ Ψµ . Thus, (Pµ )−1 = Φµ Φµ and elements of the matrix which determines the operator Vµ = (Pµ )−1 are calculated by the formula ∑

min(k,j) µ Vk,j =

φµ (k − p)φµ (j − p),

k, j ≥ 1.

p=1

The following theorem holds true. Theorem 3.3 Let functions (23) admit the canonical factorizations (24) and (25) respectively. In this case the inverse operator P−1 µ to the operator Pµ is calculated by the ′ −1 formula Pµ = Φµ Φµ , where linear operator Φµ in the space ℓ2 is determined by matrix with elements (Φµ )k,j = φµ (k − j) if 1 ≤ j ≤ k and (Φµ )k,j = 0 if j < k , k, j ≥ 1. Example 3.1 Consider an ARIMA(0,1,1) sequence {ξ(m), m ∈ Z} whose first order increments are stationary and increments with step µ = 1 form an one-sided moving average sequence of order 1 with parameter ϕ. The sequence ξ(m) has the spectral density f (λ) =

λ2 |1 − ϕe−iλ |2 . |1 − e−iλ |2

Let {η(m), m ∈ Z} be an uncorrelated with ξ(m) white noise stochastic sequence with mean 0 and variance 1. The stochastic sequence {ξ(m) + η(m), m ∈ Z} is an ARIMA(0,1,1) sequence with the spectral density f (λ) + λ2 g(λ) =

where

xλ2 |1 − ye−iλ |2 , |1 − e−iλ |2

√ 1 (3 + ϕ2 ∓ (ϕ2 − 1)2 + (ϕ − 1)2 ), 2 √ 1 y= (3 + ϕ2 ± (ϕ2 − 1)2 + (ϕ − 1)2 ). 2ϕ + 2 x=

Consider the problem of finding the mean square optimal linear estimate of the functional A1 ξ = aξ(0) + bξ(−1) which depends on unknown values ξ(0), ξ(−1) of the sequence ξ(m) from observations of the sequence ξ(m) + η(m) at points m = 0, −1, −2, . . .. Condition (12) holds true if |y| < 1. To calculate b1 ξ of the functional A1 ξ the spectral characteristic of the optimal estimate A we use formula (17). The operator Pµ = P is determined by the matrix with yp elements (P)l,k = , where p = |k − l|, l, k ≥ 1. The inverse operator x(1 − y 2 ) V = P−1 is defined by the matrix with elements (V)1,1 = x, (V)l,l = x(1 + y 2 ) Stat., Optim. Inf. Comput. Vol. 2, September 2014.


189

if l ≥ 2, (V)l,k = −xy if |l − k| = 1, l, k ≥ 1, and (V)l,k = 0 otherwise. The y l+m operator Sµ,N = S is defined by the matrix with elements (S)l,m = x(1−y 2 ) if l ≥ 1, m = −1, 0, 1, and (S)l,m = 0 if l ≥ 1, m ≥ 2. The spectral characteristic b1 ξ is calculated by the formula h1,1 (λ) of the estimate A ∞ 1 − e−iλ ∑ h1,1 (λ) = s(k)e−iλk , iλ k=0

where

s(0) = x−1 (a + b(y − 1)), s(k) = x−1 y k−1 (ay + b(y 2 − y + 1)),

k ≥ 1.

b1 ξ of the functional A1 ξ is calculated by the formula The optimal estimate A b1 ξ = a(ξ(0) + η(0)) + b(ξ(−1) + η(−1)) − A

∞ ∑

s(k)(ξ (1) (−k, 1) + η (1) (−k, 1))

k=0

= (a − s(0))(ξ(0) + η(0)) + (b + s(0) − s(1))(ξ(−1) + η(−1)) −

∞ ∑

(s(k) − s(k − 1))(ξ(−k) + η(−k))

k=2

= x−1 (a(x − 1) − b(y − 1))(ξ(0) + η(0)) + x−1 (a(1 − y) − b(y 2 − 2y − x + 2)) (ξ(−1) + η(−1)) −

∞ ∑

x−1 y k−2 (y − 1)(ay + b(y 2 − y + 1))(ξ(−k) + η(−k)).

k=2

b of the value ξ(0) is the following In particular, the optimal estimate ξ(0) b = (1 − x−1 )(ξ(0) + η(0)) − ξ(0)

∞ ∑

x−1 y k−1 (y − 1)(ξ(−k) + η(−k)).

k=1

4. Filtering of cointegrated sequences Let {ξ(m), m ∈ Z} and {ζ(m), m ∈ Z} be two integrated stochastic sequences of the same order n which define stationary nth increment sequences ξ (n) (m, µ) and ζ (n) (m, µ) with absolutely continuous spectral functions F (λ) and P (λ) which have spectral densities f (λ) and p(λ) correspondingly. Definition 4.1 Two integrated stochastic sequences {ξ(m), m ∈ Z} and {ζ(m), m ∈ Z} are called cointegrated if there exists a constant β ̸= 0 such that the sequence {ζ(m) − βξ(m) : m ∈ Z} is stationary. Stat., Optim. Inf. Comput. Vol. 2, September 2014.

190


Consider the filtering problem for cointegrated stochastic sequences which consists in finding the mean-square optimal linear estimates of the functionals Aξ =

∞ ∑

a(k)ξ(−k),

AN ξ =

k=0

N ∑

a(k)ξ(−k)

k=0

of unknown values of the sequence ξ(m) from observations of the sequence ζ(m) at points m = 0, −1, −2, . . .. This problem can be solved by using results presented in the preceding section under the condition that sequences ξ(m) and ζ(m) − βξ(m) are uncorrelated. Let the following condition holds true: ∫ π λ2n dλ < ∞. (26) iλµ |2n p(λ) −π |1 − e Determine operators Pβµ , Sβµ , Qβ with the help of the Fourier coefficients of functions λ2n |1 −

eiλµ |2n p(λ)

,

(p(λ) − β 2 f (λ)) , |1 − eiλµ |2n p(λ)

f (λ)p(λ) − β 2 f 2 (λ) λ2n p(λ)

(27)

in the same way as operators Pµ , Sµ , Q are determined in Section 3. It follows from Theorem 3.1 that the spectral characteristic hβµ (λ) of the optimal estimate ∫ π b hβµ (λ)dZζ (n) (λ), (28) Aξ = Aζ − −π

of the functional Aξ is calculated by the formula ∑∞ 2 (−iλ)n k=1 ((Pβµ )−1 Sβµ aµ )k eiλk −iλ p(λ) − β f (λ) β hµ (λ) = A(e ) − . (iλ)n p(λ) (1 − eiλµ )n p(λ)

(29)

The mean-square error of the estimate is calculated by the formula b ∆(f, g; Aξ)

β2 − 2π

+

1 2π

π

2 ∑∞ β β A(e−iλ )(1 − eiλµ )n β 2 f (λ) + λ2n k=1 ((Pµ )−1 Sµ aµ )k eiλk λ2n |1 − eiλµ |2n p2 (λ)

p(λ)dλ

−π π

2 ∑∞ β β A(e−iλ )(1 − eiλµ )n β 2 f (λ) + λ2n k=1 ((Pµ )−1 Sµ aµ )k eiλk λ2n |1 − eiλµ |2n p2 (λ)

f (λ)dλ

−π

∫

1 = 2π

∫

∫

π

2 ∑∞ β β A(e−iλ )(1 − eiλµ )n (p(λ) − β 2 f (λ)) − λ2n k=1 ((Pµ )−1 Sµ aµ )k eiλk

−π

λ2n |1 − eiλµ |2n p2 (λ) β −1 β f (λ)dλ = ⟨Sβ Sµ aµ ⟩ + ⟨Qβ a, a⟩, µ aµ , (Pµ )

(30)

In summary we have the following statement. Stat., Optim. Inf. Comput. Vol. 2, September 2014.


191

Theorem 4.1 Let the cointegrated stochastic sequences {ξ(m), m ∈ Z} and {ζ(m), m ∈ Z} have absolutely continuous spectral functions F (λ) and G(λ) which have spectral densities f (λ) and p(λ) satisfying conditions (26). Let coefficients {aµ (k) : k ≥ −µn} determined by formula (13) satisfy conditions (8). If the sequences ξ(m) b of the and ζ(m) − βξ(m) are uncorrelated, then the optimal linear estimate Aξ functional Aξ of unknown elements ξ(m), m ≤ 0, from observations of the sequence ζ(m) at points m = 0, −1, −2, . . . is calculated by formula (28). The b is calculated by formula spectral characteristic hβµ (λ) of the optimal estimate Aξ b is calculated by formula (30). (29). The value of the mean-square error ∆(f, g; Aξ) Let operators Pβµ , Sβµ,N , QβN be determined by the Fourier coefficients of functions (27) in the same way as operators Pµ , Sµ,N , QN are determined in Section 3. It follows from Theorem 3.2 that the spectral characteristic hβµ,N (λ) of the optimal estimate ∫ π b AN ξ = AN ζ − hβµ,N (λ)dZζ (n) (λ) (31) −π

of the functional AN ξ is calculated by the formula hβµ,N (λ) = AN (e−iλ )

n p(λ) − β 2 f (λ) (−iλ) − n (iλ) p(λ)

∑∞

β −1 β Sµ,N aµ,N )k eiλk k=1 ((Pµ ) . iλµ n (1 − e ) p(λ)

bN ξ is calculated by the formula The mean-square error of the estimate A

(32)

bN ξ) ∆(f, g; A

β2 − 2π 1 + 2π

π

2 ∑∞ β β AN (e−iλ )(1 − eiλµ )n β 2 f (λ) + λ2n k=1 ((Pµ )−1 Sµ,N aµ,N )k eiλk

−π

λ2n |1 − eiλµ |2n p2 (λ)

∫

1 = 2π

∫

∫

π

2 ∑∞ β β AN (e−iλ )(1 − eiλµ )n β 2 f (λ) + λ2n k=1 ((Pµ )−1 Sµ,N aµ,N )k eiλk

−π

λ2n |1 − eiλµ |2n p2 (λ)

p(λ)dλ

f (λ)dλ

π

2 ∑∞ β β AN (e−iλ )(1 − eiλµ )n (p(λ) − β 2 f (λ)) − λ2n k=1 ((Pµ )−1 Sµ,N aµ,N )k eiλk

−π

λ2n |1 − eiλµ |2n p2 (λ) β −1 β f (λ)dλ = ⟨Sβ Sµ,N aµ,N ⟩ + ⟨Qβ µ,N aµ,N , (Pµ ) N aN , aN ⟩.

(33)

The following theorem holds true. Theorem 4.2 Let the cointegrated stochastic sequences {ξ(m), m ∈ Z} and {ζ(m), m ∈ Z} have absolutely continuous spectral functions F (λ) and G(λ) which have spectral Stat., Optim. Inf. Comput. Vol. 2, September 2014.

192


densities f (λ) and p(λ) satisfying condition (26). If the sequences ξ(m) and bN ξ of the ζ(m) − βξ(m) are uncorrelated, then the optimal linear estimate A functional AN ξ of unknown elements ξ(m), −N ≤ m ≤ 0, from observations of the sequence ζ(m) at points m = 0, −1, −2, . . . is calculated by formula (31). bN ξ is calculated by The spectral characteristic hβµ,N (λ) of the optimal estimate A bN ξ) is calculated by formula (32). The value of the mean-square error ∆(f, g; A formula (33).

5. Minimax-robust method of filtering b The values of mean-square errors ∆(hµ (f, g); f, g) := ∆(f, g; Aξ) and bN ξ) and spectral characteristics hµ (f, g) ∆(hµ,N (f, g); f, g) := ∆(f, g; A b and A bN ξ of the functionals and hµ,N (f, g) of the optimal linear estimates Aξ Aξ and AN ξ of unknown values of the sequence ξ(m) based on observations of the stochastic sequence ξ(k) + η(k) are derived by formulas (15), (14) and (19), (17) correspondingly under the condition that spectral densities f (λ) and g(λ) of stochastic sequences ξ(m) and η(m) are known. In the case where spectral densities are not exactly known, but a set D = Df × Dg of admissible spectral densities is given, the minimax (robust) approach to estimation of functionals which depend on the unknown values of stochastic sequence with stationary increments is reasonable. In other words we are interested in finding an estimate that minimizes the maximum of mean-square errors for all spectral densities from a given class D = Df × Dg of admissible spectral densities simultaneously. Definition 5.1 For a given class of spectral densities D = Df × Dg spectral densities f0 (λ) ∈ Df , g0 (λ) ∈ Dg are called least favorable in the class D for the optimal linear filtering of the functional Aξ if the following relation holds true ∆(f0 , g0 ) = ∆(h(f0 , g0 ); f0 , g0 ) =

max

(f,g)∈Df ×Dg

∆(h(f, g); f, g).

Definition 5.2 For a given class of spectral densities D = Df × Dg the spectral characteristic h0 (λ) of the optimal linear estimate of the functional Aξ is called minimax-robust if there are satisfied conditions ∩ h0 (λ) ∈ HD = L02 (f (λ) + λ2n g(λ)), (f,g)∈Df ×Dg

min

max

h∈HD (f,g)∈Df ×Dg

∆(h; f, g) =

max

(f,g)∈Df ×Dg

∆(h0 ; f, g).



193

Using the derived in the previous sections formulas and the introduced definitions we can conclude that the following statement holds true. Lemma 5.1 Spectral densities f 0 ∈ Df , g 0 ∈ Dg which satisfy condition (12) are least favorable in the class D = Df × Dg for the optimal linear filtering of the functional Aξ if operators P0µ , S0µ , Q0 determined by the Fourier coefficients of the functions λ2n |1 − eiλµ |2n (f 0 (λ) + λ2n g 0 (λ))

,

λ2n g 0 (λ) |1 − eiλµ |2n (f 0 (λ) + λ2n g 0 (λ))

,

f 0 (λ)g 0 (λ) f 0 (λ) + λ2n g 0 (λ)

determine a solution of the conditional extremum problem 0 0 −1 0 max(⟨Sµ aµ , P−1 Sµ aµ ⟩ + ⟨Q0 a, a⟩. (34) µ Sµ aµ ⟩ + ⟨Qa, a⟩) = ⟨Sµ aµ , (Pµ ) f ∈D

The minimax spectral characteristic is determined as h0 = hµ (f 0 , g 0 ) if hµ (f 0 , g 0 ) ∈ HD . The function h0 and the pair (f 0 , g 0 ) form a saddle point of the function ∆(h; f, g) on the set HD × D. The saddle point inequalities ∆(h; f 0 , g 0 ) ≥ ∆(h0 ; f 0 , g 0 ) ≥ ∆(h0 ; f, g)

∀f ∈ Df , ∀g ∈ Dg , ∀h ∈ HD

hold true if h0 = hµ (f 0 , g 0 ) and hµ (f 0 , g 0 ) ∈ HD , where (f 0 , g 0 ) is a solution of the conditional extremum problem: e g) = −∆(hµ (f 0 , g 0 ); f, g) → inf, ∆(f,

(f, g) ∈ D,

∆(hµ (f 0 , g 0 ); f, g)

=

+

1 2π

1 2π

π

2 ∑∞ A(e−iλ )(1 − eiλµ )n f 0 (λ) + λ2n k=1 ((P0µ )−1 S0µ aµ )k eiλk

−π

|1 − eiλµ |2n (f 0 (λ) + λ2n g 0 (λ))2

∫

g(λ)dλ

π

2 ∑∞ A(e−iλ )(1 − eiλµ )n λ2n g 0 (λ) − λ2n k=1 ((P0µ )−1 S0µ aµ )k eiλk

−π

λ2n |1 − eiλµ |2n (f 0 (λ) + λ2n g 0 (λ))2

∫

f (λ)dλ.

This conditional extremum problem is equivalent to the unconditional extremum problem e g) + δ(f, g|Df × Dg ) → inf, ∆D (f, g) = ∆(f, where δ(f, g|Df × Dg ) is the indicator function of the set Df × Dg . Solution (f 0 , g 0 ) to this unconditional extremum problem is characterized by condition 0 ∈ ∂∆D (f 0 , g 0 ) (see [24, 34]).


194


6. Least favorable spectral densities in the class Df0 × Dg0 Consider the problem of minimax filtering of the functional Aξ for the set of admissible spectral densities D = Df0 × Dg0 , where { } { } ∫ π ∫ π 1 1 Df0 = f (λ)| f (λ)dλ ≤ P1 , Dg0 = g(λ)| g(λ)dλ ≤ P2 . 2π −π 2π −π Let us assume that densities f 0 ∈ Df0 , g 0 ∈ Dg0 and functions hµ,f (f 0 , g 0 ) =

∑∞ A(e−iλ )(1 − eiλµ )n λ2n g 0 (λ) − λ2n k=1 ((P0µ )−1 S0µ aµ )k eiλk |λ|n |1 − eiλµ |n (f 0 (λ) + λ2n g 0 (λ))

,

(35) iλk −iλ iλµ n 0 2n ∑∞ 0 −1 0 Sµ aµ )k e )(1 − e ) f (λ) + λ A(e k=1 ((Pµ ) hµ,g (f 0 , g 0 ) = (36) iλµ n 0 2n 0 |1 − e

| (f (λ) + λ 0

g (λ))

0

are bounded. In this case the functional ∆(hµ (f , g ); f, g) is continuous and bounded in the L1 × L1 space. It comes from the condition 0 ∈ ∂∆D (f 0 , g 0 ) that least favorable densities f 0 ∈ Df0 , g 0 ∈ Dg0 satisfy the equation ∞ ∑ −iλ iλµ n 2n 0 2n 0 −1 0 iλk ((Pµ ) Sµ aµ )k e A(e )(1 − e ) λ g (λ) − λ k=1

= α1 |λ|n |1 − eiλµ |n (f 0 (λ) + λ2n g 0 (λ)),

∞ ∑ iλk −iλ iλµ n 0 2n 0 −1 0 ((Pµ ) Sµ aµ )k e A(e )(1 − e ) f (λ) + λ

(37)

k=1

= α2 |1 − e

| (f (λ) + λ2n g 0 (λ)),

iλµ n

0

(38)

1 ∫π 0 where α1 ≥ 0 and α2 ≥ 0 are constants such that α1 ̸= 0 if f (λ)dλ = P1 2π −π 1 ∫π 0 and α2 ̸= 0 if g (λ)dλ = P2 . 2π −π Thus, we have the following statements.

Theorem 6.1 Let spectral densities f 0 (λ) ∈ Df0 and g 0 (λ) ∈ Dg0 satisfy condition (12), let functions hµ,f (f 0 , g 0 ) and hµ,g (f 0 , g 0 ) be bounded. The spectral densities f 0 (λ) and g 0 (λ) determined by equations (37), (38) are least favorable in the class D = Df0 × Dg0 for the optimal linear estimation of the functional Aξ if they determine solution of extremum problem (34). The function hµ (f 0 , g 0 ) determined by formula (14) is minimax spectral characteristic of the optimal estimate of the functional Aξ . Stat., Optim. Inf. Comput. Vol. 2, September 2014.


195

Theorem 6.2 Let spectral density f (λ) be known, let spectral density g 0 (λ) ∈ Dg0 and let conditions (12) be satisfied. Let the function hµ,g (f, g 0 ) be bounded. Spectral density g 0 (λ) is least favorable in the class Dg0 for the optimal linear filtering of the functional Aξ if it is of the form {

1 g (λ) = 2n max λ 0

0,

A(e−iλ )(1 − eiλµ )n f (λ) + λ2n ∑∞ ((P0µ )−1 S0µ aµ )k eiλk k=1 α2 |1 − eiλµ |n

}

− f (λ)

and the pair (f, g 0 ) determines a solution of the extremum problem (34). The function hµ (f, g 0 ) determined by formula (14) is minimax spectral characteristic of the optimal estimation of the functional Aξ . Theorem 6.3 Let spectral density g(λ) be known, let spectral density f 0 (λ) ∈ Df0 and let condition (12) be satisfied. Let the function hµ,f (f 0 , g) be bounded. Spectral density f 0 (λ) is least favorable in the class Df0 for the optimal linear filtering of the functional Aξ if it is of the form {

f 0 (λ) = max

0,

|λ|n A(e−iλ )(1 − eiλµ )n g(λ) − α1 |1 −

∑∞

}

0 −1 S0 a ) eiλk µ µ k k=1 ((Pµ ) iλµ n e |

− λ2n g(λ)

and the pair (f 0 , g) determines a solution of the extremum problem (34). The function hµ (f 0 , g) determined by formula (14) is minimax spectral characteristic of the optimal estimation of the functional Aξ . v 7. Least favorable densities in the class D = Du × Dε

Consider the problem of optimal linear filtering of the functional Aξ for the set of spectral densities D = Duv × Dε , where } { ∫ π 1 v f (λ)dλ ≤ P1 , Du = f (λ)|v(λ) ≤ f (λ) ≤ u(λ), 2π −π { } ∫ π 1 Dε = g(λ)|g(λ) = (1 − ε)g1 (λ) + εw(λ), g(λ)dλ ≤ P2 . 2π −π Here spectral densities u(λ), v(λ), g1 (λ) are known and fixed, and spectral densities u(λ), v(λ) are bounded. Let spectral densities f 0 ∈ Duv , g 0 ∈ Dε be such that functions hµ,f (f 0 , g 0 ) and hµ,g (f 0 , g 0 ) determined by formulas (35), (36) are bounded. From the condition 0 ∈ ∂∆D (f 0 , g 0 ) we find the following equations that define least favorable densities ∞ ∑ ((P0µ )−1 S0µ aµ )k eiλk A(e−iλ )(1 − eiλµ )n λ2n g 0 (λ) − λ2n k=1


196


= α1 |λ|n |1 − eiλµ |n (f 0 (λ) + λ2n g 0 (λ))(γ1 (λ) + γ2 (λ) + α1−1 ), ∞ ∑ ((P0µ )−1 S0µ aµ )k eiλk A(e−iλ )(1 − eiλµ )n f 0 (λ) + λ2n

(39)

k=1

= α2 |1 − e

| (f (λ) + λ g (λ))(φ(λ) + α2−1 ),

iλµ n

0

2n 0

(40)

where γ1 ≤ 0 and γ1 = 0 if f 0 (λ) ≥ v(λ); γ2 (λ) ≥ 0 and γ2 = 0 if f 0 (λ) ≤ u(λ); φ(λ) ≤ 0 and φ(λ) = 0 when g 0 (λ) ≥ (1 − ε)g1 (λ). The following statements hold true. Theorem 7.1 Let spectral densities f 0 (λ) ∈ Duv , g 0 (λ) ∈ Dε satisfy condition (12). Let functions hµ,f (f 0 , g 0 ) and hµ,g (f 0 , g 0 ) determined by formulas (35), (36) be bounded. Spectral densities f 0 (λ) and g 0 (λ) determined by equations (39), (40) are least favorable in the class D = Duv × Dε for the optimal linear filtering of the functional Aξ if they determine a solution of extremum problem (34). The function hµ (f 0 , g 0 ) determined by (14) is minimax spectral characteristic of the optimal estimate of the functional Aξ . Theorem 7.2 Let spectral density f (λ) be known, let spectral density g 0 (λ) ∈ Dε and let condition (12) be satisfied. Assume that the function hµ,g (f, g 0 ) determined by formula (36) is bounded. Spectral density g 0 (λ) is least favorable in the class Dε for the optimal linear filtering of the functional Aξ if it is of the form 1 max {(1 − ε)g1 (λ), f1 (λ)} , λ2n A(e−iλ )(1 − eiλµ )n f (λ) + λ2n ∑∞ ((P0µ )−1 S0µ aµ )k eiλk k=1 − f (λ), f1 (λ) = α2 |1 − eiλµ |n g 0 (λ) =

and the pair (f, g 0 ) determines a solution of extremum problem (34). The function hµ (f, g 0 ) determined by formula (14) is minimax spectral characteristic of the optimal estimate of the functional Aξ . Theorem 7.3 Let spectral density g(λ) be known, let spectral density f 0 (λ) ∈ Duv and let condition (12) be satisfied. Let the function hµ,f (f 0 , g) be bounded. Spectral density f 0 (λ) is least favorable in the class Duv for the optimal linear filtering of the functional Aξ if it is of the form f 0 (λ) = min {v(λ), max {u(λ), g2 (λ)}} , ∑∞ |λ|n A(e−iλ )(1 − eiλµ )n g(λ) − k=1 ((P0µ )−1 S0µ aµ )k eiλk g2 (λ) = − λ2n g(λ) α1 |1 − eiλµ |n Stat., Optim. Inf. Comput. Vol. 2, September 2014.


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and the pair (f 0 , g) determines a solution of extremum problem (34). The function hµ (f 0 , g) determined by formula (14) is minimax spectral characteristic of the optimal estimation of the functional Aξ .

8. Conclusions In this article we propose solutions of the filtering problem for functionals Aξ and AN ξ which depend on unobserved values of a stochastic sequence ξ(k) with stationary nth increments. Estimates are based on observations of the sequence ξ(k) + η(k) at points of time k = 0, −1, −2, . . . , where η(k) is an uncorrelated with ξ(k) stationary sequence. We derived formulas for calculating values of the mean-square errors and spectral characteristics of the optimal linear estimates of the functionals in the case where spectral densities of the sequences are known. The obtained results are applied to find solution of the filtering problem for cointegrated sequences. In the case of spectral uncertainty where spectral densities are not known exactly, but a set of admissible spectral densities is specified, the minimax-robust method is applied. Formulas that determine least favorable spectral densities and minimax (robust) spectral characteristics are derived for some special sets of admissible spectral densities. The filtering problem for ARIMA(0,1,1) sequence is analyzed as an example of application of the developed method.

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