VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA51 10.15559/16-VMSTA51 Research Article Minimax interpolation of sequences with stationary increments and cointegrated sequences LuzMaksymmaksim_luz@ukr.net MoklyachukMikhailmmp@univ.kiev.ua∗ Taras Shevchenko National University of Kyiv, Kyiv, Ukraine Corresponding author. 2016 142016315978 1132016 1632016 1732016 © 2016 The Author(s). Published by VTeX2016 Open access article under the CC BY license.

We consider the problem of optimal estimation of the linear functional ANξ=∑k=0Na(k)ξ(k) depending on the unknown values of a stochastic sequence ξ(m) with stationary increments from observations of the sequence ξ(m)+η(m) at points of the set Z∖{0,1,2,…,N} , where η(m) is a stationary sequence uncorrelated with ξ(m) . We propose formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional in the case of spectral certainty, where spectral densities of the sequences are exactly known. We also consider the problem for a class of cointegrated sequences. We propose relations that determine the least favorable spectral densities and the minimax spectral characteristics in the case of spectral uncertainty, where spectral densities are not exactly known while a set of admissible spectral densities is specified.

 Stochastic sequence with stationary increments cointegrated sequences minimax-robust estimate mean square error least favorable spectral density minimax-robust spectral characteristic 60G10 60G25 60G35 62M20 93E10 93E11 Introduction

In this paper, we investigate the problem of estimating the missed observations of stochastic sequences with stationary increments. Kolmogorov [13], Wiener [27], and Yaglom [29, 30] developed effective methods of estimation of the unknown values of stationary sequences and processes. Later on Yaglom [28] and Pinsker [21] introduced and investigated stochastic processes with stationary increments of order n. Properties of these and other processes generalizing the concept of stationarity are described in the books by Yaglom [29, 30]. The stationary and related stochastic sequences are widely used in econometrics and in financial time series analysis. Examples of these sequences are autoregressive sequences (AR), moving-average sequences (MA), and autoregressive moving-average sequences (ARMA). Time series with trends are described by integrated ARMA sequences (ARIMA) and seasonal time series, which are examples of stochastic sequences with stationary increments. These models are properly described in the book by Box, Jenkins, and Reinsel [2]. Granger [8] introduced a concept of cointegrated sequences, namely, the integrated sequences such that some linear combination of them has a lower order of integration. Cointegrated sequences are described in more details in the paper by Engle and Granger [5]. We also refer to the papers [3, 4, 9, 12] for recent developments.

Traditional methods of finding solutions to extrapolation, interpolation, and filtering problems for stationary and related stochastic processes are developed under the basic assumption that the spectral densities of the considered stochastic processes are exactly known. However, in most practical situations, complete information on the spectral densities of the processes is not available. Investigators can apply the traditional methods considering the estimated spectral densities instead of the true ones. However, as it was shown by Vastola and Poor [26] with the help of some examples, this approach can result in significant increasing of the value of the error of estimate. Therefore, it is reasonable to derive estimates that are optimal for all densities from a certain class of spectral densities. These estimates are called minimax-robust since they minimize the maximum of the mean-square errors for all spectral densities from a set of admissible spectral densities simultaneously. This approach to study the problem of extrapolation of stationary stochastic processes was introduced by Grenander [10]. Franke [6] investigated the minimax extrapolation and interpolation problems for stationary sequences applying the convex optimization methods. In the book by Moklyachuk [20], the minimax-robust estimates of the linear functionals of stationary sequences and processes are presented. See also the survey paper [18], The classical and minimax-robust problems of interpolation, extrapolation, and filtering of the functional of stochastic sequences with stationary increments are investigated in the papers by Luz and Moklyachuk [14–17, 19]. Particularly, the cointegrated sequences are investigated in the papers [14, 15]. The classical extrapolation problem in the case where both the signal and the noise processes are not stationary was investigated by Bell [1].

In the present paper, we consider the problem of estimation of the linear functional ANξ= ∑k=0Na(k)ξ(k), which depends on the unknown values of the sequence ξ(k) with stationary nth increments based on observations of the sequence ξ(k)+η(k) at points m∈Z∖{0,1,2,…,N} . The sequence η(k) is assumed to be stationary and uncorrelated with ξ(k) .

 Stationary increment stochastic sequences. Spectral representation

In this section, we present the main results of the spectral theory of stochastic sequences with nth stationary increments. For more details, we refer to the books by Yaglom [29, 30].

For a given stochastic sequence {ξ(m),m∈Z} , the sequence ξ(n)(m,μ)=(1−Bμ)nξ(m)= ∑l=0n(−1)lnlξ(m−lμ), where Bμ is the backward shift operator with step μ∈Z such that Bμξ(m)=ξ(m−μ) , is called a stochastic nth increment sequence with step μ∈Z .

The stochastic nth increment sequence ξ(n)(m,μ) generated by a 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 the mean value of the nth increment sequence, and the function D(n)(m,μ1,μ2) is called the structural function of the stationary nth increment sequence (or the structural function of nth order of the stochastic sequence {ξ(m),m∈Z} ).

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, D(n)(m,μ1,μ2)=∫−ππeiλm(1−e−iμ1λ)n(1−eiμ2λ)n1λ2ndF(λ), 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) .

Representation (3) and the Karhunen theorem [7] give us a spectral representation of the stationary nth increment sequence ξ(n)(m,μ) : ξ(n)(m,μ)=∫−ππeimλ(1−e−iμλ)n1(iλ)ndZξ(n)(λ), where Zξ(n)(λ) is a random process with uncorrelated increments on [−π,π) with respect to the spectral function F(λ) : E|Zξ(n)(t2)−Zξ(n)(t1)|2=F(t2)−F(t1)∀−π≤t1<t2<π.

We will use the spectral representation (4) for deriving the optimal linear estimates of unknown values of stochastic sequences with stationary increments.

 Hilbert space projection method of interpolation

Consider a stochastic sequence {ξ(m),m∈Z} with stationary nth increments ξ(n)(m,μ) and uncorrelated with ξ(m) stationary stochastic sequence {η(m),m∈Z} . Suppose that these sequences have absolutely continuous spectral functions F(λ) and G(λ) with spectral densities f(λ) and g(λ) , respectively. We will suppose that the stationary increment ξ(n)(m,μ) and the stationary sequence η(m) have zero mean values and μ>0 0$]]>.

Interpolation problem for the sequences ξ(m) and η(m) is considered as the problem of the mean-square optimal estimation of the linear functional ANξ= ∑k=0Na(k)ξ(k), which depends on the unknown values of the stochastic sequence ξ(m) at points m=0,1,…,N based on observations of the sequence ζ(m)=ξ(m)+η(m) at points of the set Z∖{0,1,2…,N} .

Suppose that the spectral densities f(λ) and g(λ) satisfy the minimality condition ∫−ππλ2n|1−eiλμ|2n(f(λ)+λ2ng(λ))dλ<∞. Under this condition, the mean-square error of the estimate of the functional ANξ is not equal to zero [24].

The functional ANξ admits the representation ANξ=ANζ−ANη=BNζ−ANη−VNζ=HNξ−VNζ, where HNξ:=BNζ−ANη,ANζ= ∑k=0Na(k)ζ(k),ANη= ∑k=0Na(k)η(k), BNζ= ∑k=0Nbμ,N(k)ζ(n)(k,μ),VNζ= ∑k=−μn−1vμ,N(k)ζ(k). The coefficients vμ,N(k) , k=−μn,−μn+1,…,−1 , and bμ,N(k) , k=0,1,2,…,N , are calculated by the formulas (see [15]) vμ,N(k)= ∑l=[−kμ]′min{[N−kμ],n}(−1)lnlbμ,N(lμ+k),k=−μn,−μn+1,…,−1, bμ,N(k)= ∑m=kNa(m)dμ(m−k)=(DNμaN)k,k=0,1,…,N, where by [x]′ we denote the least integer number among the numbers that are greater than or equal to x, the coefficients {dμ(k):k≥0} are determined by the relationship ∑k=0∞dμ(k)xk=( ∑j=0∞xμj)n, the matrix DNμ of dimension (N+1)×(N+1) is defined by the coefficients (DNμ)k,j=dμ(j−k) if 0≤k≤j≤N , and (DNμ)k,j=0 if 0≤j<k≤N ; and aN=(a(0),a(1),a(2),…,a(N))′ is a vector of dimension (N+1) .

The functional HNξ from representation (6) has finite variance, and the functional VNζ depends on the known observations of the stochastic sequence ζ(k) at the points k=−μn,−μn+1,…,−1 . Therefore, optimal estimates AˆNξ and HˆNξ of the functionals ANξ and HNξ and the mean-square errors Δ(f,g;AˆNξ)=E|ANξ−AˆNξ|2 and Δ(f,g;HˆNξ)=E|HNξ−HˆNξ|2 of the estimates AˆNξ and HˆNξ satisfy the following relations: AˆNξ=HˆNξ−VNζ,Δ(f,g;AˆNξ)=E|ANξ−AˆNξ|2=E|HNξ−VNζ−HˆNξ+VNζ|2=E|HNξ−HˆNξ|2=Δ(f,g;HˆNξ). Thus, the interpolation problem for the functional ANξ is equivalent to the interpolation problem for the functional HNξ . This problem can be solved by applying the Hilbert space projection method proposed by Kolmogorov [13]. The optimal linear estimate AˆNξ of the functional ANξ can be represented in the form AˆNξ=∫−ππhμ(λ)dZξ(n)+η(n)(λ)− ∑k=−μn−1vμ,N(k)(ξ(k)+η(k)), where hμ(λ) is the spectral characteristic of the optimal estimate HˆNξ .

Let H0−(ξμ(n)+ημ(n)) be the closed linear subspace generated by elements {ξ(n)(k,μ)+η(n)(k,μ):k≤−1} of the Hilbert space H=L2(Ω,F,P) of random variables γ with zero mean value and finite variance, Eγ=0 , E|γ|2<∞ , with the inner product (γ1;γ2)=Eγ1γ2‾ . Let HN+(ξ−μ(n)+η−μ(n)) be the closed linear subspace of the Hilbert space H=L2(Ω,F,P) generated by elements {ξ(n)(k,−μ)+η(n)(k,−μ):k≥N+1} . The equality ξ(n)(k,−μ)=(−1)nξ(n)(k+μn,μ) implies HN+(ξ−μ(n)+η−μ(n))=H(N+μn)+(ξμ(n)+ημ(n)). Let us also define the subspaces L20−(p) and L2N+(p) of the Hilbert space L2(p) with the inner product (x1;x2)=∫−ππx1(λ)x2(λ)‾p(λ)dλ that are generated by the functions {eiλk(1−e−iλμ)n(iλ)−n:k≤−1} and {eiλk(1−e−iλμ)n(iλ)−n:k≥N+1} , respectively, where the function p(λ)=f(λ)+λ2ng(λ) is the spectral density of the sequence ζ(m) , m∈Z [15]. The optimal estimate HˆNξ of the functional HNξ is the projection of the element HNξ of the Hilbert space H=L2(Ω,F,P) onto the subspace H0−(ξμ(n)+ημ(n))⊕HN+(ξ−μ(n)+η−μ(n))=H0−(ξμ(n)+ημ(n))⊕H(N+μn)+(ξμ(n)+ημ(n)). The following conditions characterize the estimate HˆNξ :

HˆNξ∈H0−(ξμ(n)+ημ(n))⊕H(N+μn)+(ξμ(n)+ημ(n)) ;

(HNξ−HˆNξ)⊥H0−(ξμ(n)+ημ(n))⊕H(N+μn)+(ξμ(n)+ημ(n)) .

 The functional HNξ in the space H admits the spectral representation HNξ=∫−ππBNμ(eiλ)(1−e−iλμ)n(iλ)ndZξ(n)+η(n)(λ)−∫−ππAN(eiλ)dZη(λ), BNμ(eiλ)= ∑k=0Nbμ,N(k)eiλk= ∑k=0N(DNμaN)keiλk,AN(eiλ)= ∑k=0Na(k)eiλk. Making use of the described representation and condition 2), we derive the following equation for determining the spectral characteristic hμ(λ) : ∫−ππ[(BNμ(eiλ)(1−e−iλμ)n(iλ)n−hμ(λ))p(λ)−A(eiλ)g(λ)(−iλ)n]×(1−eiλμ)n(−iλ)ne−iλkdλ=0∀k≤−1,∀k≥N+μn+1. Thus, the spectral characteristic hμ(λ) can be represented as follows: hμ(λ)=BNμ(eiλ)(1−e−iλμ)n(iλ)n−AN(eiλ)(−iλ)ng(λ)p(λ)−(−iλ)nCNμ(eiλ)(1−eiλμ)np(λ),CNμ(eiλ)= ∑k=0N+μncμ(k)eiλk, where cμ(k) , k=0,1,2,…,N+μn , are unknown coefficients we have to determine. Condition 1) implies that the spectral characteristic hμ(λ) satisfies the following equations: ∫−ππBNμ(eiλ)−AN(eiλ)λ2ng(λ)(1−e−iλμ)np(λ)−λ2nCNμ(eiλ)|1−eiλμ|2np(λ)e−iλldλ=0,0≤l≤N+μn. The derived equations are represented as a system of N+μn+1 linear equations: bμ,N(l)− ∑m=0N+μnTl,mμaμ,N(m)= ∑k=0N+μnPl,kμcμ(k),0≤l≤N, − ∑m=0N+μnTl,mμaμ,N(m)= ∑k=0N+μnPl,kμcμ(k),N+1≤l≤N+μn, where the coefficients {aμ,N(m):0≤m≤N+μn} are calculated by the formula aμ,N(m)= ∑l=max{[m−Nμ]′,0}min{[mμ],n}(−1)lnla(m−μl),0≤m≤N+μn, and the Fourier coefficients {Tk,jμ,Pk,jμ:0≤k,j≤N+μn} are calculated by the formulas Tk,jμ=12π∫−ππeiλ(j−k)λ2ng(λ)|1−eiλμ|2n(f(λ)+λ2ng(λ))dλ,0≤k,j≤N+μn, Pk,jμ=12π∫−ππeiλ(j−k)λ2n|1−eiλμ|2n(f(λ)+λ2ng(λ))dλ,0≤k,j≤N+μn.

Denote by [DNμaN]+μn the vector of dimension (N+μn+1) constructed by adding μn zeros to the vector DNμaN of dimension (N+1) . Using these definitions, system (12)–(13) can be represented in the matrix form [DNμaN]+μn−TNμaNμ=PNμcNμ, where aNμ=(aμ,N(0),aμ,N(1),aμ,N(2),…,aμ,N(N+μn))′ and cNμ=(cμ(0),cμ(1),cμ(2),…,cμ(N+μn))′ are vectors of dimension (N+μn+1) ; and PNμ and TNμ are matrices of dimension (N+μn+1)×(N+μn+1) with elements (PNμ)l,k=Pl,kμ and (TNμ)l,k=Tl,kμ , 0≤l,k≤N+μn . Thus, the coefficients cμ(k) , 0≤k≤N+μn , are determined by the formula cμ(k)=((PNμ)−1[DNμaN]+μn−(PNμ)−1TNμaμ)k,0≤k≤N+μn, where ((PNμ)−1[DNμaN]+μn−(PNμ)−1TNμaNμ)k , 0≤k≤N+μn , is the kth element of the vector (PNμ)−1[DNμaN]+μn−(PNμ)−1TNμaNμ . The existence of the invertible matrix (PNμ)−1 was shown in [25] under condition (5). The spectral characteristic hμ(λ) of the estimate HˆNξ of the functional HNξ is calculated by formula (11), where CNμ(eiλ)= ∑k=0N+μn((PNμ)−1[DNμaN]+μn−(PNμ)−1TNμaNμ)keiλk. The value of the mean-square errors of the estimates AˆNξ and HˆNξ can be calculated by the formula Δ(f,g;AˆNξ)=Δ(f,g;HˆNξ)=E|HNξ−HˆNξ|2=12π∫−ππ|AN(eiλ)(1−eiλμ)nf(λ)−λ2nCNμ(eiλ)|2|1−eiλμ|2n(f(λ)+λ2ng(λ))2g(λ)dλ+12π∫−ππ|AN(eiλ)(1−eiλμ)nλ2ng(λ)+λ2nCNμ(eiλ)|2λ2n|1−eiλμ|2n(f(λ)+λ2ng(λ))2f(λ)dλ=⟨[DNμaN]+μn−TNμaNμ,(PNμ)−1[DNμaN]+μn−(PNμ)−1TNμaNμ⟩+⟨QNaN,aN⟩, where QN is the matrix of dimension (N+1)×(N+1) with the coefficients (QN)l,k=Ql,k , 0≤l,k≤N , calculated by the formula Qk,j=12π∫−ππeiλ(j−k)f(λ)g(λ)f(λ)+λ2ng(λ)dλ,0≤k,j≤N.

We can summarize the derived results in the form of the following theorem.

Let {ξ(m),m∈Z} be a stochastic sequence with stationary nth increments ξ(n)(m,μ) , and let {η(m),m∈Z} be a stationary stochastic sequence uncorrelated with ξ(m) . Let the spectral densities f(λ) and g(λ) of the sequences satisfy the minimality condition (5). The optimal linear estimate AˆNξ of the functional ANξ , which depends on the values ξ(m) , 0≤m≤N , based on the observations of the sequence ξ(m)+η(m) at points of the set Z∖{0,1,2,…,N} is calculated by formula (10). The spectral characteristic hμ(λ) and the value of the mean-square error Δ(f,g;AˆNξ) of the optimal estimate AˆNξ are calculated by formulas (11) and (15), respectively.

Let the spectral density f(λ) of the sequence ξ(m) satisfy the minimality condition ∫−ππλ2n|1−eiλμ|2nf(λ)dλ<∞. The optimal linear estimate AˆNξ of the functional ANξ of unknown values ξ(m) , 0≤m≤N , based on observations of the sequence ξ(m) at the points m∈Z∖{0,1,2,…,N} can be calculated by the formula AˆNξ=∫−ππhμξ(λ)dZξ(n)(λ)− ∑k=−μn−1vμ,N(k)ξ(k). The spectral characteristic hμξ(λ) and the mean-square error Δ(f;AˆNξ) of the optimal estimate AˆNξ can be calculated by the formulas hμξ(λ)=BNμ(eiλ)(1−e−iλμ)n(iλ)n−(−iλ)n∑k=0N+μn((FNμ)−1[DNμaN]+μn)keiλk(1−eiλμ)nf(λ), Δ(f;AˆNξ)=12π∫−ππλ2n|∑k=0N+μn((FNμ)−1[DNμaN]+μn)keiλk|2|1−eiλμ|2nf(λ)dλ=⟨(FNμ)−1[DNμaN]+μn,[DNμaN]+μn⟩, where FNμ is the matrix of dimension (N+μn+1)×(N+μn+1) with elements (FNμ)k,j=Fk,jμ , 0≤k,j≤N+μn , Fk,jμ=12π∫−ππeiλ(j−k)λ2n|1−eiλμ|2nf(λ)dλ,0≤k,j≤N+μn.

In the case of estimation of an unobserved value ξ(p) , 0≤p≤N , the following statement holds true.

Let the conditions of Theorem 2 hold. The optimal linear estimate ξˆ(p) of an unobserved value ξ(p) , 0≤p≤N , of the stochastic sequence with nth stationary increments based on observations of the sequence ξ(m)+η(m) at the points m∈Z∖{0,1,2,…,N} is calculated by the formula ξˆ(p)=∫−ππhμ,p(λ)dZξ(n)+η(n)(λ)− ∑l=1n(−1)lnl(ξ(p−μl)+η(p−μl)), hμ,p(λ)=(1−e−iλμ)n(iλ)n ∑k=0pdμ(p−k)eiλk−eiλp(−iλ)ng(λ)p(λ)−(−iλ)nCpμ(eiλ)(1−eiλμ)np(λ), Cpμ(eiλ)= ∑k=0N+μn((PNμ)−1dμ,p−(PNμ)−1Tpμan)keiλk, where dμ,p=(dμ(p),dμ(p−1),dμ(p−2),…,dμ(0),0,…,0)′ and an=(an(0),an(1),…,an(n),0,…,0)′, an(k)=(−1)knk , k=0,1,2,…,n , are vectors of dimension (N+μn+1) , Tpμ is the (N+μn+1)×(N+μn+1) matrix with elements (Tpμ)l,k=Tl,p+μkμ if 0≤l≤N+μn , 0≤k≤n , and (Tpμ)l,k=0 if 0≤l≤N+μn , N+1≤k≤N+μn . The value of the mean-square error of the optimal estimate is calculated the by formula Δ(f,g;ξˆ(p))=⟨dμ,p−Tpμan,(PNμ)−1dμ,p−(PNμ)−1Tpμan⟩+Q0,0.

In the case of estimating the sequence ξ(m) with nth stationary increments at points of the set Z∖{0,1,2,…,N} , the optimal linear estimate of a value ξ(p) , 0≤p≤N , is calculated by the formula ξˆ(p)=∫−ππhμ,pξ(λ)dZξ(n)(λ)− ∑l=1n(−1)lnlξ(p−μl), hμ,pξ(λ)=(1−e−iλμ)n(iλ)n ∑k=0pdμ(p−k)eiλk−(−iλ)n∑k=0N+μn((FNμ)−1dμ,p)keiλk(1−eiλμ)nf(λ). The value of the mean-square error of the estimate is calculated by the formula Δ(f;ξˆ(p))=12π∫−ππλ2n|∑k=0N+μn((FNμ)−1dμ,p)keiλk|2|1−eiλμ|2nf2(λ)dλ=⟨(FNμ)−1dμ,p,dμ,p⟩.

Consider the stochastic sequence ξ(m) , m∈Z , defined by the equation ξ(m)=(1−ϕ)ξ(m−1)+ϕξ(m−2)+ε(m), which means that values of the sequence ξ(m) are defined as a weighted sum of two previous values of the sequence plus a value ε(m) of the sequence of independent identically distributed random variables with mean value Eε(m)=0 and variance Eε2(m)=1 .

Consider the increment ξ(1)(m;1)=ξ(m)−ξ(m−1) of the sequence. We can find that ξ(1)(m;1)=−ϕξ(1)(m−1;1)+ε(m). Thus, the increment sequence ξ(1)(m;1) with step μ=1 is an autoregressive sequence with parameter 0<ϕ<1 . The sequence ξ(m) is an ARIMA(1;1;0) sequence with the spectral density f(λ)=λ2|1−e−iλ|2|1+ϕe−iλ|2.

Let us find the estimate Aˆ1ξ of the value of the functional A1ξ=2ξ(0)+ξ(1) based on observations of the sequence ξ(m) at the points m∈Z∖{0,1} . Let ϕ=1/2 . In this case, v1,1(−1)=−2 , F1=14520252025,F1−1=48521−104−1025−104−1021,[D11a1]+1=310. Therefore, h1ξ(λ)=−10685e−iλ−485e3iλ, Aˆ1ξ=−10685ξ(1)(−1;1)−485ξ(1)(3;1)−3ξ(−1)=10685ξ(−2)+14985ξ(−1)+485ξ(2)−485ξ(3). The value of the mean-square error of the estimate is Δ(f,g;Aˆ1ξ)=8817 .

 Interpolation of cointegrated sequences

Consider two integrated sequences {ξ(m),m∈Z} and {ζ(m),m∈Z} with absolutely continuous spectral functions F(λ) and P(λ) and the corresponding spectral densities f(λ) and p(λ) .

Two integrated sequences {(ξ(m),ζ(m)),m∈Z} are called cointegrated (of order 0) if, for some constant β≠0 , the linear combination {ζ(m)−βξ(m):m∈Z} is a stationary sequence.

The interpolation problem for cointegrated sequences consists in mean-square optimal linear estimation of the functional ANξ= ∑k=0Na(k)ξ(k) of unknown values of the stochastic sequence ξ(m) based on observations of the stochastic sequence ζ(m) at the points m∈Z∖{0,1,2,…,N} . To solve the problem, we can use the results obtained in the previous sections.

Suppose that the spectral density p(λ) of the sequence ζ(m) satisfies the minimality condition ∫−ππλ2n|1−eiλμ|2np(λ)dλ<∞. Let the matrices PNμ,β , TNμ,β , QNβ be defined by the Fourier coefficients of the functions λ2n|1−eiλμ|2np(λ),p(λ)−β2f(λ)|1−eiλμ|2np(λ),[f(λ)p(λ)−β2f2(λ)]+λ2np(λ) in the same way as the matrices PNμ , TNμ , QN were defined. Theorem 2 implies the following formula for calculating the spectral characteristic hμ,Nβ(λ) of the optimal estimate AˆNξ=∫−ππhμ,Nβ(λ)dZζ(n)(λ)− ∑k=−μn−1vμ,N(k)ζ(k) of the functional ANξ : hμ,Nβ(λ)=BNμ(eiλ)(1−e−iλμ)n(iλ)n−AN(eiλ)p(λ)−β2f(λ)(iλ)np(λ)−(−iλ)nCμ,Nβ(eiλ)(1−eiλμ)np(λ), where Cμ,Nβ(eiλ)= ∑k=0N+μn((PNμ,β)−1[DNμaN]+μn−(PNμ,β)−1TNμ,βaNμ)keiλk.

The value of the mean-square error of the estimate AˆNξ is calculated by the formula Δ(f,g;AˆNξ)=12π∫−ππ|AN(eiλ)(1−eiλμ)nβ2f(λ)−λ2nCμ,Nβ(eiλ)|2λ2n|1−eiλμ|2np2(λ)p(λ)dλ−β22π∫−ππ|AN(eiλ)(1−eiλμ)nβ2f(λ)−λ2nCμ,Nβ(eiλ)|2λ2n|1−eiλμ|2np2(λ)f(λ)dλ+12π∫−ππ|AN(eiλ)(1−eiλμ)n[p(λ)−β2f(λ)]++λ2nCμ,Nβ(eiλ)|2λ2n|1−eiλμ|2np2(λ)f(λ)dλ=⟨[DNμaN]+μn−TNμ,βaNμ,(PNμ,β)−1[DNμaN]+μn−(PNμ,β)−1TNμ,βaNμ⟩+⟨QNβaN,aN⟩.

The described results are presented as the following theorem.

Let {(ξ(m),ζ(m)),m∈Z} be two cointegrated sequences with spectral densities f(λ) and p(λ) , and let the spectral density p(λ) satisfy the minimality condition (19). If the stochastic sequences ξ(m) and ζ(m)−βξ(m) are uncorrelated, then the spectral characteristic hμ,Nβ(λ) and the value of the mean-square error Δ(f,g;AˆNξ) of the optimal estimate AˆNξ (21) of the functional ANξ based on the observations of the sequence ζ(m) at the points m∈Z∖{0,1,2,…,N} are calculated by formulas (22) and (23), respectively.

 Minimax-robust method of interpolation

Formulas for calculating values of the mean-square error Δ(h(f,g);f,g)=Δ(f,g;AˆNξ)=E|ANξ−AˆNξ|2 and the spectral characteristics of the optimal estimates of the functional ANξ based on observations of the sequence ξ(m)+η(m) can be applied under the condition that the spectral densities f(λ) and g(λ) of the stochastic sequences ξ(m) and η(m) are known. However, these formulas often cannot be used in many practical situations since the exact values of the densities are not available. In this situation, the minimax-robust method can be applied. It consists in finding the estimate that provides a minimum of the mean-square errors for all spectral densities from a given set D=Df×Dg of admissible spectral densities simultaneously.

For a given class of spectral densities D=Df×Dg , spectral densities f0(λ)∈Df and g0(λ)∈Dg are called the least favorable densities in the class D for the optimal linear interpolation of the functional ANξ if the following relation holds: Δ(f0,g0)=Δ(h(f0,g0);f0,g0)=max(f,g)∈Df×DgΔ(h(f,g);f,g).

For a given class of spectral densities D=Df×Dg , the spectral characteristic h0(λ) of the optimal linear estimate of the functional ANξ is called minimax-robust if the following conditions are satisfied: h0(λ)∈HD=⋂(f,g)∈Df×DgL20−(p)⊕L2(N+μn)+(p), minh∈HDmax(f,g)∈Df×DgΔ(h;f,g)=max(f,g)∈Df×DgΔ(h0;f,g).

The spectral densities f0∈Df and g0∈Dg that satisfy the minimality condition (5) are the least favorable in the class D for the optimal linear interpolation of the functional ANξ based on observations of the sequence ξ(m)+η(m) at the points m∈Z∖{0,1,2,…,N} if the matrices (PNμ)0 , (TNμ)0 , (QN)0 whose elements are defined by the Fourier coefficients of the functions λ2n|1−eiλμ|2np0(λ),λ2ng0(λ)|1−eiλμ|2np0(λ),f0(λ)g0(λ)p0(λ), where p0(λ)=f0(λ)+λ2ng0(λ) , determine a solution to the constrained optimization problem max(f,g)∈Df×Dg(⟨[DNμaN]+μn−TNμaμ,(PNμ)−1[DNμaN]+μn−(PNμ)−1TNμaNμ⟩+⟨QNaN,aN⟩)=⟨[DNμaN]+μn−(TNμ)0aNμ,((PNμ)0)−1[DNμaN]+μn−((PNμ)0)−1(TNμ)0aNμ⟩+⟨QN0aN,aN⟩. The minimax-robust spectral characteristic h0=hμ(f0,g0) is calculated by formula (11) if hμ(f0,g0)∈HD .

The presented statements follow from the introduced definitions and Theorem 2.

The minimax-robust spectral characteristic h0 and the least favorable spectral densities (f0,g0) form a saddle point of the function Δ(h;f,g) on the set HD×D . The saddle-point inequalities Δ(h;f0,g0)≥Δ(h0;f0,g0)≥Δ(h0;f,g)∀f∈Df,∀g∈Dg,∀h∈HD hold if h0=hμ(f0,g0) , hμ(f0,g0)∈HD , and (f0,g0) is a solution to the constrained optimization problem Δ˜(f,g)=−Δ(hμ(f0,g0);f,g)→inf,(f,g)∈D, Δ(hμ(f0,g0);f,g)=12π∫−ππ|AN(eiλ)(1−eiλμ)nf0(λ)−λ2nCNμ,0(eiλ)|2|1−eiλμ|2n(f0(λ)+λ2ng0(λ))2g(λ)dλ+12π∫−ππ|AN(eiλ)(1−eiλμ)nλ2ng0(λ)+λ2nCNμ,0(eiλ)|2λ2n|1−eiλμ|2n(f0(λ)+λ2ng0(λ))2f(λ)dλ, CNμ,0(eiλ)= ∑k=0N+μn(((PNμ)0)−1[DNμaN]+μn−((PNμ)0)−1(TNμ)0aNμ)keiλk. This constrained optimization problem is equivalent to the unconstrained optimization problem ΔD(f,g)=Δ˜(f,g)+Δ(f,g|Df×Dg)→inf, where Δ(f,g|Df×Dg) is the indicator function of the set Df×Dg : Δ(f,g|Df×Dg)=0 if (f;g)∈Df×Dg and Δ(f,g|Df×Dg)=+∞ if (f;g)∉Df×Dg .

A solution (f0,g0) to the unconstrained optimization problem is determined by the condition 0∈∂ΔD(f0,g0) , which is a necessary and sufficient condition that the pair (f0,g0) belongs to the set of minimums of the convex functional ΔD(f,g) [11, 22, 23]. By ∂ΔD(f,g) we denote the subdifferential of the functional ΔD(f,g) at the point (f,g)=(f0,g0) , that is, the set of all linear continuous functionals Λ on the space L1×L1 that satisfy the inequality ΔD(f,g)−ΔD(f0,g0)≥Λ((f,g)−(f0,g0)),(f,g)∈D.

In the case of estimating the cointegrated sequences, we have the following optimization problem of finding the least favorable spectral densities: ΔD(f,p)=Δ˜(f,p)+Δ(f,p|Df×Dp)→inf, Δ˜(f,p)=Δ(hμβ(f0,p0);f,p)=12π∫−ππ|AN(eiλ)(1−eiλμ)nβ2f0(λ)−λ2nCμ,Nβ,0(eiλ)|2λ2n|1−eiλμ|2n(p0(λ))2p(λ)dλ−β22π∫−ππ|AN(eiλ)(1−eiλμ)nβ2f0(λ)−λ2nCμ,Nβ,0(eiλ)|2λ2n|1−eiλμ|2nl(p0(λ))2f(λ)dλ+12π∫−ππ|AN(eiλ)(1−eiλμ)n[p0(λ)−β2f0(λ)]++λ2nCμ,Nβ,0(eiλ)|2λ2n|1−eiλμ|2n(p0(λ))2f(λ)dλ Cμ,Nβ,0(eiλ)= ∑k=0N+μn(((PNμ,β)0)−1([DNμaN]+μn−(TNμ,β)0aNμ))keiλk. A solution (f0,p0) to this optimization problem is characterized by the condition 0∈∂ΔD(f0,p0) .

The derived representations of the linear functionals Δ(hμ(f0,g0);f,g) and Δ(hμβ(f0,p0);f,p) allow us to calculate derivatives and subdifferentials in the space L1×L1 . Therefore, the complexity of the optimization problems (26) and (27) is determined by the complexity of calculation of the subdifferentials of the indicator functions Δ(f,g|Df×Dg) and Δ(f,p|Df×Dp) of the sets Df×Dg and Df×Dp .

 The least favorable spectral densities in the class <inline-formula id="j_vmsta51_ineq_312"><alternatives> <mml:math><mml:msubsup><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo mathvariant="normal">,</mml:mo><mml:mi mathvariant="italic">f</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msubsup><mml:mo>×</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo mathvariant="normal">,</mml:mo><mml:mi mathvariant="italic">g</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msubsup></mml:math> <tex-math><![CDATA[${\mathcal{D}_{0,f}^{-}}\times {\mathcal{D}_{0,g}^{-}}$]]></tex-math></alternatives></inline-formula>

Consider the problem of minimax-robust estimation of the functional ANξ of unknown values of the sequence with stationary increments ξ(m) based on observations of the sequence ξ(m)+η(m) at the points m∈Z∖{0,1,2,…,N} for the set of admissible spectral densities D=D0,f−×D0,g− , where D0,f−=f(λ)|12π∫−ππ1f(λ)dλ≥P1, D0,g−=g(λ)|12π∫−ππ1g(λ)dλ≥P2. If the spectral densities f0∈D0,f− , g0∈D0,g− and the functions hμ,f(f0,g0)=|AN(eiλ)(1−eiλμ)nλ2ng0(λ)+λ2nCNμ,0(eiλ)||λ|n|1−eiλμ|np0(λ), hμ,g(f0,g0)=|AN(eiλ)(1−eiλμ)nf0(λ)−λ2nCNμ,0(eiλ)||1−eiλμ|np0(λ), where p0(λ)=f0(λ)+λ2ng0(λ) , are bounded, then the linear functional Δ(hμ(f0,g0);f,g) is continuous and bounded in the space L1×L1 . The condition 0∈∂ΔD(f0,g0) implies that the spectral densities f0∈D0,f− and g0∈D0,g− are determined by the relations |λ|nf0(λ)|AN(eiλ)(1−eiλμ)ng0(λ)+CNμ,0(eiλ)|=α1|1−eiλμ|n(f0(λ)+λ2ng0(λ)), g0(λ)|AN(eiλ)(1−eiλμ)nf0(λ)−λ2nCNμ,0(eiλ)|=α2|1−eiλμ|n(f0(λ)+λ2ng0(λ)), where the constants α1≥0 , α2≥0 with α1≠0 if ∫−ππ(f0(λ))−1dλ=2πP1 and α2≠0 if ∫−ππ(g0(λ))−1dλ=2πP2 .

The derived statements allow us to formulate the following theorems.

Suppose that the spectral densities f0(λ)∈D0,f− and g0(λ)∈D0,g− satisfy the minimality condition (5) and the functions hμ,f(f0,g0) and hμ,g(f0,g0) calculated by formulas (28) and (29) are bounded. The spectral densities f0(λ) and g0(λ) determined by Eqs. (30)) and (31) are the least favorable densities in the class D=D0,f−×D0,g− for the linear interpolation of the functional ANξ if they give a solution to the constrained optimization problem (25). The function hμ(f0,g0) calculated by formula (11) is the minimax-robust spectral characteristic of the optimal estimate of the functional ANξ .

Suppose that the spectral density f(λ) (or g(λ)) is known, the spectral density g0(λ)∈D0,g− (f0(λ)∈D0,f−) , and they satisfy the minimality condition (5). Suppose also that the function hμ,g(f,g0) (hμ,f(f0,g)) is bounded. Then the spectral density g0(λ)=f(λ)1α2|1−eiλμ|n|AN(eiλ)(1−eiλμ)nf(λ)−CNμ,0(eiλ)|−λ2n+−1 or f0(λ)=λ2ng(λ)|λ|nα1|1−eiλμ|n|AN(eiλ)(1−eiλμ)ng(λ)+CNμ,0(eiλ)|−1+−1 is the least favorable in the class D0,g− (or D0,f−) for the linear interpolation of the functional ANξ if the functions f(λ)+λ2ng0(λ) , g0(λ) (or f0(λ)+λ2ng(λ)) give a solution to the constrained optimization problem (25). The function hμ(f,g0) (or hμ(f0,g)) calculated by formula (11) is the minimax-robust spectral characteristic of the optimal estimate of the functional ANξ .

Consider the problem of minimax-robust estimation of the functional ANξ of unknown values of the sequence ξ(m) , cointegrated with the sequence ζ(m) , based on observations of the sequence ζ(m) at the points m∈Z∖{0,1,2,…,N} . Suppose that the stochastic sequences ξ(m) and ζ(m)−βξ(m) are uncorrelated. The least favorable spectral densities in the class Df0×Dp0 , where D0,f−=f(λ)|12π∫−ππ1f(λ)dλ≥P1,D0,p−=p(λ)|12π∫−ππ1p(λ)dλ≥P2, are determined by the condition 0∈∂ΔD(f0,p0) , which implies the following relations for determining the least favorable spectral densities f0∈Df0 and p0∈Dp0 : |AN(eiλ)(1−eiλμ)nβ2f0(λ)−λ2nCμ,Nβ,0(eiλ)|=α2|λ|n|1−eiλμ|n, f0(λ)|AN(eiλ)(1−eiλμ)n[p0(λ)−β2f0(λ)]++λ2nCμ,Nβ,0(eiλ)|=|λ|n|1−eiλμ|n(α1p0(λ)+α2|β|f0(λ)), where the constants α1≥0 , α2≥0 with α1≠0 if ∫−ππ(f0(λ))−1dλ=2πP1 and α2≠0 if ∫−ππ(p0(λ))−1dλ=2πP2 .

Suppose that the spectral density p0(λ)∈D0,p− satisfies the minimality condition (19) and the functions hμ,f(f0,g0) and hμ,g(f0,g0) , calculated by formulas (28) and (29), are bounded for g(λ):=λ−2n(p(λ)−β2f(λ)) . The spectral densities f0(λ) and p0(λ) determined by Eqs. (32) and (33) are the least favorable in the class D=D0,f−×D0,p− for the linear interpolation of the functional ANξ based on observations of the stochastic sequence ζ(m) , which is cointegrated with ξ(m) and such that the stochastic sequences ξ(m) and ζ(m)−βξ(m) are uncorrelated, if these densities determine a solution to constrained optimization problem (25) for g0(λ):=λ−2n(p0(λ)−β2f0(λ)) . The function hμ(f0,p0) , calculated by formula (22), is the minimax-robust spectral characteristic of the optimal estimate of the functional ANξ .

 The least favorable spectral densities in the class <inline-formula id="j_vmsta51_ineq_389"><alternatives> <mml:math><mml:mi mathvariant="script">D</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant="italic">ε</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="script">D</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant="italic">ε</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math> <tex-math><![CDATA[$\mathcal{D}=\mathcal{D}_{2\varepsilon _{1}}\times \mathcal{D}_{1\varepsilon _{2}}$]]></tex-math></alternatives></inline-formula>

Consider the problem of minimax-robust interpolation of the functional ANξ based on observations of the sequence ξ(m)+η(m) at the points of m∈Z∖{0,1,2,…,N} in the case where the spectral densities f(λ) and g(λ) belong to the set D=D2ε1×D1ε2 , where D2ε1=f(λ)|12π∫−ππ|f(λ)−f1(λ)|2dλ≤ε1, D1ε2=g(λ)|12π∫−ππ|g(λ)−g1(λ)|dλ≤ε2 are ε-neighborhoods of the given spectral densities f1(λ) and g1(λ) in the spaces L2 and L1 , respectively.

Suppose that the spectral densities f1(λ) and g1(λ) are bounded and the functions hμ,f(f0,g0) and hμ,g(f0,g0) calculated by formulas (28) and (29) with spectral densities f0∈D2ε1 and g0∈D1ε2 are bounded as well. The condition 0∈∂ΔD(f0,g0) implies the following relations for determining the least favorable spectral densities: |AN(eiλ)(1−eiλμ)nλ2ng0(λ)+λ2nCNμ,0(eiλ)|2=α1|λ|2n|1−eiλμ|2n(f0(λ)−f1(λ))(f0(λ)+λ2ng0(λ))2, |AN(eiλ)(1−eiλμ)nf0(λ)−λ2nCNμ,0(eiλ)|2=α2γ(λ)|1−eiλμ|2n(f0(λ)+λ2ng0(λ))2, where the function |γ(λ)|≤1 and γ(λ)=sign(g(λ)−g1(λ)) if g(λ)≠g1(λ) ; α1 ,  α2 are two constants to be found using the equations 12π∫−ππ|f0(λ)−f1(λ)|2dλ=ε1,12π∫−ππ|g0(λ)−g1(λ)|dλ=ε2.

Now we can present the following theorems, which describe the least favorable spectral densities in the class D=D2ε1×D1ε2 .

Suppose that the spectral densities f0(λ)∈D2ε1 and g0(λ)∈D1ε2 satisfy the minimality condition (5), the functions hμ,f(f0,g0) and hμ,g(f0,g0) , calculated by formulas (28) and (29), are bounded. The spectral densities f0(λ) and g0(λ) determined by equations (34)–(36) are the least favorable spectral densities in the class D=D2ε1×D1ε2 for the linear interpolation of the functional ANξ if they give a solution to constrained optimization problem (25). The function hμ(f0,g0) , calculated by formula (11) is the minimax-robust spectral characteristic of the optimal estimate of the functional ANξ .

Suppose that the spectral density f(λ) is known, the spectral density g0(λ)∈D1ε2 , and they satisfy the minimality condition (5). Suppose also that the function hμ,g(f,g0) calculated by formula (29) is bounded. Then the spectral density g0(λ)=max{g1(λ),λ−2nf2(λ)}, f2(λ)=α2−1|1−eiλμ|−nAN(eiλ)(1−eiλμ)nf(λ)−λ2nCNμ,0(eiλ)−f(λ), is the least favorable in the class D1ε2 for the linear interpolation of the functional ANξ if a pair (f,g0) provides a solution to constrained optimization problem (25). The function hμ(f,g0) , calculated by formula (11) is the minimax-robust spectral characteristic of the optimal estimate of the functional ANξ .

Suppose that the spectral density g(λ) is known, the spectral density f0(λ)∈D2ε1 , and they satisfy the minimality condition (5). Suppose also that the function hμ,f(f0,g) , calculated by formula (28), is bounded. The spectral density f0(λ) determined by the equation |AN(eiλ)(1−eiλμ)nλ2ng(λ)+λ2nCNμ,0(eiλ)|2=α1|λ|2n|1−eiλμ|2n(f0(λ)−f1(λ))(f0(λ)+λ2ng(λ))2 and the condition ∫−ππ|f0(λ)−f1(λ)|2dλ=2πε1 is the least favorable spectral density in the class D2ε1 for the linear interpolation of the functional ANξ if a pair (f0,g) provides a solution to constrained optimization problem (25). The function hμ(f0,g) calculated by formula (11) is the minimax-robust spectral characteristic of the optimal estimate of the functional ANξ .

Consider the problem of minimax-robust interpolation of the functional ANξ in the case of cointegrated sequences ξ(m) and ζ(m) on the set of admissible spectral densities D=D2ε1×D1ε2 , where D2ε1=f(λ)|12π∫−ππ|f(λ)−f1(λ)|2dλ≤ε1, D1ε2=p(λ)|12π∫−ππ|p(λ)−p1(λ)|dλ≤ε2. From the condition 0∈∂ΔD(f0,g0) we obtain the following relations that determine the least favorable spectral densities: |AN(eiλ)(1−eiλμ)nβ2f0(λ)−λ2nCμ,Nβ,0(eiλ)|2=α2λ2nγ(λ)|1−eiλμ|2n(p0(λ))2, |AN(eiλ)(1−eiλμ)n[p0(λ)−β2f0(λ)]++λ2nCμ,Nβ,0(eiλ)|2=λ2n|1−eiλμ|2n(p0(λ))2(α1(f0(λ)−f1(λ))+α2β2γ(λ)), where the function |γ(λ)|≤1 and γ(λ)=sign(p(λ)−p1(λ)) if p(λ)≠p1(λ) ; α1 ,  α2 are two constants that can be found from the equations 12π∫−ππ|f0(λ)−f1(λ)|2dλ=ε1,12π∫−ππ|p0(λ)−p1(λ)|dλ=ε2.

Thus, we have the following theorem.

Suppose that the spectral density p0(λ)∈D1ε2 satisfies the minimality condition (19) and the functions hμ,f(f0,g0) and hμ,g(f0,g0) , calculated by formulas (28) and (29), are bounded for g(λ):=λ−2n(p(λ)−β2f(λ)) . Then the least favorable spectral densities for the linear interpolation of the functional ANξ based on observations of the stochastic sequence ζ(m) , which is cointegrated with ξ(m) and such that the stochastic sequences ξ(m) and ζ(m)−βξ(m) are uncorrelated, are the spectral densities f0(λ) and p0(λ) determined by Eqs. (37)–(39) and provide a solution to constrained optimization problem (25) for g0(λ):=λ−2n(p0(λ)−β2f0(λ)) . The function hμ(f0,p0) calculated by formula (22) is the minimax-robust spectral characteristic of the optimal estimate of the functional ANξ .

 Conclusions

In the article, the problem of the mean-square optimal linear estimation of the functional ANξ=∑k=0Na(k)ξ(k) , which depends of unknown values of the sequence ξ(m) with nth stationary increments based on observations of the sequence ξ(m)+η(m) at the points m∈Z∖{0,1,2,…,N} , is considered in the case of observations with the stationary noise η(m) uncorrelated with ξ(m) . The classical and minimax-robust methods of interpolation are applied in the case of spectral certainty and in the case spectral uncertainty. Particularly, in the case of spectral certainty, formulas for calculating the spectral characteristics and the value of the mean-square error of the optimal estimate are found. The derived results are applied to interpolation problem for a class of cointegrated sequences. In the case spectral uncertainty, where spectral densities are not known exactly, whereas some sets of admissible spectral densities are given, formulas that determine the least favorable spectral densities and the minimax-robust spectral characteristics are derived for some special sets of admissible spectral densities.

 Acknowledgments

The authors would like to thank the referee for careful reading of the article and giving constructive suggestions.

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