Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences

The problem of optimal estimation of the linear functionals Aξ = ∑∞ k=0 a(k)ξ(k) and AN ξ = ∑N k=0 a(k)ξ(k) which depend on the unknown values of a stochastic sequence ξ(m) with stationary nth increments is considered. Estimates are obtained which are based on observations of the sequence ξ(m) + η(m) at points of time m = −1,−2, . . ., where the sequence η(m) is stationary and uncorrelated with the sequence ξ(m). Formulas for calculating the mean-square errors and the spectral characteristics of the optimal estimates of the functionals are derived in the case of spectral certainty, where spectral densities of the sequences ξ(m) and η(m) are exactly known. These results are applied for solving extrapolation problem for cointegrated sequences. In the case where spectral densities of the sequences are not known exactly, but sets of admissible spectral densities are given, the minimax-robust method of estimation is applied. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special classes of admissible densities.


Introduction
The theory of estimation of the unknown values of stationary processes based on a set of observations plays an important role in many practical applications.The development of the theory started from the classical works of Kolmogorov [18] and Wiener [42], in which they presented methods of solution of the extrapolation and interpolation problems for stationary processes.The interpolation problem considered by Kolmogorov means estimation of the missed values of a stochastic sequence.The prediction problem consists in estimation the future values of the process based on observations of the process in the past.The third classical problem is filtering of random processes which consists in estimation the original values of the signal process from observations of the process with noise.All these problems for stationary sequences and processes are clearly described in the book by Rozanov [41].Most of results which have appeared since that time were based on the assumption that the spectral structure of the stationary process is known.After the main points of the new theory were established, scientists tried to generalize the concept of stationarity.One of the natural generalization was proposed by Yaglom [45], Pinsker [38], Yaglom and Pinsker [37].They proposed a class of processes with stationary increments of nth order.
They described basic properties of these processes, found the spectral representation of the stationary increment and solved the extrapolation problem for processes with stationary nth increments.Other generalizations of the concept of stationarity can be found in the books by Yaglom [43,44].
One of the fields of practical applications of the stationary and related stochastic sequences is economical modeling and financial time series.Most simple examples of stationary linear models are moving average (MA) sequences, autoregressive (AR) and autoregressive-moving average (ARMA) sequences, state space models, all of which refer to stationary sequences with rational spectral function without unit AR-roots.Models with trends and seasonal components are represented by integrated ARMA (ARIMA) sequences and seasonal time series.The spectral structure of these sequences has unit roots in the autoregressive part.These sequences are most simple examples of sequences with stationary increments.Such models have been inducing the interest of scientists for the last 30 years.The main results concerning the model description, parameter estimation, forecasting and further investigations are described in the classical book by Box, Jenkins and Reinsel [2].Statistical investigations of real data found some specific relations between integrated sequences.In some cases linear combinations of such sequences appears to be stationary.Such property Grander [5] called cointegration.Cointegrated models found their application in applied and theoretical econometrics and financial time series [11].
As we have already mentioned the problem of estimation of unobserved values of the investigated time series is important in mathematical studies.The classical methods of extrapolation, interpolation and filtering relay on the exact information about the spectral densities of the investigated processes.However, in practise none of the methods of estimation can provide the exact representation of spectral structure of the process.In the case where spectral densities are not known exactly, but a set of admissible spectral densities are given, we can apply the minimax (robust) method of estimation, which allows us to determine estimates that minimize the value of mean-square error for all densities from a given class.Grenander [12] was the first one who applied this approach to the extrapolation problem for stationary processes.In the papers by Franke [13], Franke and Poor [14], Kassam and Poor [17] the minimax extrapolation and interpolation problem for stationary sequences was solved by using convex optimization techniques.In the works by Moklyachuk [27] - [33] problems of extrapolation, interpolation and filtering for stationary processes and sequences were studied.The minimax extrapolation problem for functionals which depend on the unknown values of stationary sequences from observations with noise is solved in the paper by Moklyachuk [26].The corresponding problems for vector-valued stationary sequences and processes were investigated by Moklyachuk and Masyutka [30] - [35].In the articles by Dubovets'ka and Moklyachuk [6] - [10] and the book by Golichenko and Moklyachuk [3] the minimax estimation problems were investigated for another generalization of stationary processes -periodically correlated stochastic sequences and random processes.
Luz and Moklyachuk investigated the classical and minimax extrapolation, interpolation and filtering problems for sequences and processes with nth stationary increments.They presented solutions of the filtering problem for the linear functionals Aξ = ∑ ∞ k=0 a(k)ξ(−k) and A N ξ = ∑ N k=0 a(k)ξ(−k) in the papers [21,23].The minimax interpolation problem for the linear functional A N ξ = ∑ N k=0 a(k)ξ(k) which depends on the unknown values of the sequence ξ(k) based on observations with and without noise was investigated in papers [19,20], and for the linear functional Aξ = ∫ ∞ 0 a(t)ξ(t)dt which depends on the unknown values of a random process ξ(t) in the paper [24].
In papers by Luz and Moklyachuk [22,25] the problem of optimal linear extrapolation of linear functionals which depend on the unknown values of stochastic sequences and random processes with nth stationary increments from the observations without noise is investigate.The classical extrapolation problem for a non-stationary sequence which is observed with a non-stationary noise was studied by Bell [1].However, he showed that the problem can be solved under additional assumptions, particularly if we have an additional finite set of values of the sequence ξ(m).
In the proposed paper we consider the extrapolation problem for the functionals which depend on the unknown values of a stochastic sequence ξ(k) with stationary nth increments based on observations of the sequence ξ(k) + η(k) at points k = −1, −2, . .., where η(k) is a stationary stochastic sequence uncorrelated with the sequence ξ(k).Under the condition of stationarity of the noise η(k) we solve the problem without additional assumptions described by Bell [1].The obtained estimates give us a method of solvution the extrapolation problem for cointegrated sequences ξ(m) and ζ(m) assuming stationarity of a linear combination of these sequences.The estimation problem is also solved in the case of spectral uncertainty where spectral densities of sequences are not exactly known but a set of admissible spectral densities is given.Formulas that determine the least favorable spectral densities and the minimax-robust spectral characteristic of the optimal linear estimates of the functional Aξ are derived in the case of spectral uncertainty for some concrete classes of admissible spectral densities.

Definition 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 exist for all m 0 , µ, m, µ 1 , µ 2 and do not depend on m 0 .The function c (n) (µ) is called mean value of the nth increment sequence and the function 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 1
The mean value c (n) (µ) and the structural function Stat., Optim.Inf.Comput.Vol.On other hand, a function c (n) (µ) which has form (4) with a constant c and a function D (n) (m, µ 1 , µ 2 ) 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).
Representation (5) and the Karhunen theorem [4,16] give us the spectral representation of the stationary nth increment sequence ξ (n) (m, µ): where Z ξ (n) (λ) is a random process with independent increments on [−π, π) connected with the spectral function F (λ) by the relation Denote by H(ξ (n) ) a subspace generated in the Hilbert space Since the space S(ξ (n) ) is a subspace in the Hilbert space H(ξ (n) ), the space H(ξ (n) ) admits the decomposition where R(ξ (n) ) is the orthogonal complement of the subspace S(ξ (n) ) in the space H(ξ (n) ).

Definition 3
A stationary nth increment sequence Theorem 2 A wide-sense stationary stochastic increment sequence ξ (n) (m, µ) admits a unique representation in the form where {ξ Components of representation (8) are defined by the formulas Define by H p (ε) the Hilbert subspace generated by elements {ε(m) : m ≤ p}.We will call the sequence {ε(m) : m ∈ Z} an innovation sequence for a regular stationary nth increment sequence

Theorem 3
A stochastic stationary increment sequence ξ (n) (m, µ) is regular if and only if there exists an innovation sequence {ε(m) : m ∈ Z} and a sequence of complex functions {φ (n) Representation ( 9) is called canonical moving average representation of the stochastic stationary increment sequence ξ (n) (m, µ).

MINIMAX PREDICTION PROBLEM FOR STOCHASTIC SEQUENCES WITH STATIONARY INCREMENTS
Corollary 1 A wide-sense stationary stochastic increment sequence ξ (n) (m, µ) admits a unique representation (10) where and {ε m : m ∈ Z} is an innovation sequence.If the stationary nth increment sequence ξ (n) (m, µ) admit the canonical representation (9), then its spectral function F (λ) has the spectral density f (λ) admitting the canonical factorization where the function Φ(z) = ∑ ∞ k=0 φ(k)z k has the convergence radius r > 1 and does not have zeros in the unit disk where φ µ (k) = φ (n) (k, µ) are coefficients from the canonical representation (9).Then the following relation holds true: In the next section we will use spectral representation (6) and canonical factorization (12) for finding the optimal mean square estimate of the unknown values of the stochastic sequence {ξ(m), m ∈ Z} with nth stationary increments.

Extrapolation problem
Consider a stochastic sequence {ξ(m), m ∈ Z} which generates a stationary nth increment sequence ξ (n) (m, µ) with absolutely continuous spectral function F (λ) and 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) and spectral density g(λ).From now we will assume that mean values of the increment sequence ξ (n) (m, µ) and stationary sequence η(m) equal to 0. We will also consider the increment step µ > 0.
In this section our purpose is to solve the problem of linear mean-square optimal estimation of the functionals which depend on unknown values of the sequence ξ(m) based on observations of the sequence First of all we indicate some conditions which are necessary for solving the considered problem.Assume that coefficients a(k), k ≥ 0, and the linear transformation D µ which is defined in the following part of the section satisfy the conditions and Stat., Optim.Inf.Comput.Vol. 3, June 2015 Assume also that spectral densities f (λ) and g(λ) satisfy the minimality condition In order to find an estimate of the functional Aξ we have to formulate the extrapolation problem in terms of linear functionals of some stationary sequences.The functional Aξ can be presented as Under conditions (13) the functional Aη has finite second moment.
We will exploit the representation of the functional Aζ which is proposed in [25] and is described in the following lemma.
where [x] ′ denotes the least integer number among numbers which are greater or equal to x, {d(k) : k ≥ 0} are coefficients determined by the relation ) n , D µ is a linear operator determined by , and D µ k,j = 0 if j < k, the vector a = (a(0), a(1), a(2), . ..) ′ .
From Lemma 1 we get the following representation of the functional Aξ: where Hξ = Bζ − Aη.Denote by ∆(f, g; Aξ) = E|Aξ − Aξ| 2 the mean-square error of the optimal estimate Aξ of the functional Aξ and by ∆(f, g; Hξ) = E|Hξ − Hξ| 2 the mean-square error of the optimal estimate Hη of the functional Hη.Since the functional V ζ is determined by the observed values of ζ(k) at points k = −µn, −µn + 1, . . ., −1, the following relations hold true To find the mean-square optimal estimate of the functional Hξ we apply the Hilbert space orthogonal projection method proposed by Kolmogorov [18].The stationary stochastic sequence η(m) admits the spectral representation where Z η (λ) is a random process with independent increments on [−π, π) corresponding to the spectral function G(λ).The random processes Z η (λ) and Z η (n) (λ) are connected by the relation [23].The spectral density p(λ) of the sequence ζ(m) is determined by spectral densities f (λ) and g(λ) by the relation The functional Hξ admits the following spectral representation: where Denote by H 0− (ξ µ ) a closed linear subspace of the Hilbert space H = L 2 (Ω, F, P) of random variables having finite second moments which is generated by values {ξ (n) The representation yields a one to one correspondence between elements e iλk (1 Every linear estimate Aξ of the functional Aξ admits the representation where h µ (λ) is the spectral characteristic of the estimate Hξ.The mean square optimal estimate Hξ can be found as a projection of the element Hξ on the subspace H 0− (ξ This projection is determined by two conditions: The second condition implies the following relations which hold true for all k ≤ −1: These relations can be represented in the form which allows us to derive the spectral characteristic h µ (λ) of the estimate Hξ.It has the form where c µ (k), k ≥ 0, are unknown coefficients which we need to determine.It follows from condition 1) that the spectral characteristic h µ (λ) admits the representation where which leads to the conditions Determine for every k, j ∈ Z the Fourier coefficients of the corresponding functions dλ.
Using these Fourier coefficients we can represent equation (20) in terms of the system of linear equations where This system of equations can be written in the form where , a µ (2), . ..) ′ ; P µ and T µ are linear operators in the space ℓ 2 defined by the matrices with elements Consequently, the unknown coefficients c µ (k), k ≥ 0, which determine the spectral characteristic h µ (λ) are calculated by the formula where Thus, the spectral characteristic h µ (λ) of the optimal estimate Hξ of the functional Hξ is calculated by the formula . (22) The mean-square error of the estimate Aξ is calculated by the formula where Q is a linear operator in the space ℓ 2 defined by the matrix with elements 2), . ..) ′ , y = (y(0), y(1), y(2), . ..) ′ .These reasonings can be summarized in the form of the theorem.

Corollary 3
The spectral characteristic h µ (λ) admits the representation where Here h 1 µ (λ) and h 2 µ (λ) are the spectral characteristics of the optimal estimates Bζ and Aη of the functionals Bζ and Aη respectively based on observations ξ(k) + η(k) at points k = −1, −2, . ... Theorem 4 allows us to obtain the optimal estimate A N ξ of the functional A N ξ which depend on the unknown values of elements ξ(m), m = 0, 1, 2, . . ., N , based on observations of the sequence ξ(m) + η(m) at points m = −1, −2, . ... Put a(k) = 0, k > N .Then the spectral characteristic h µ,N (λ) of the linear estimate is calculated by the formula where T µ,N is a linear operator in the space ℓ 2 defined by the matrix with elements (T µ,N ) l,m = T µ l,m , l ≥ 0, 0 ≤ m ≤ N + µn, and (T µ,N ) l,m = 0, l ≥ 0, m > N + µn.Here [x] ′ denotes the least integer number among numbers which are greater or equal to x.The mean-square error of the estimate A N ξ is calculated by the formula where Q N is a linear operator in the space ℓ 2 defined by the matrix with elements The following theorem holds true.

Theorem 5
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 the minimality condition (15) be satisfied.The optimal linear estimate A N ξ of the functional A N ξ which depend on the unknown values of elements ξ(k), k = 0, 1, 2, . . ., N , from observations of the sequence ξ(m) + η(m) at points m = −1, −2, . . . is calculated by formula (26).The spectral characteristic h µ,N (λ) of the optimal estimate A N ξ is calculated by formula (27).The value of the mean-square error ∆(f, g; A N ξ) is calculated by formula (29).
A particular case of the considered problem is the problem of forecasting of an unobserved value of a stochastic sequence ξ(p) at point p, p ≥ 0, from observations of the sequence ξ(k) + η(k) at points k = −1, −2, . ... In this case the vector a µ,N has coefficients a µ,N (m) = (−1) l ( n l ) if m = p + µl, l = 0, 1, 2, . . ., n, m ≥ 0, and a µ,N (m) = 0 otherwise.Let us define a vector a n = (a n (0), a n (1), . . ., a n (n), 0, 0, . ..) ′ , where of the value ξ(p), p ≥ 0, can be calculated by the formula where is a linear operator in the space ℓ 2 defined by the matrix with elements The mean-square error of the estimate is calculated by the formula Thus, we have the following statement.
Theorems 4, 5 and Corollary 4 determine solutions of the extrapolation problem for the linear functionals Aξ, A N ξ and the value ξ(p), p ≥ 0, using the Fourier coefficients of the functions However, the problem of finding the inverse operator (P µ ) −1 to the operator P µ defined by the Fourier coefficients of the function is complicated in most cases.
Fortunately, the proposed formulas can be simplified under the assumption that the functions , g(λ) (33) admit the canonical factorizations Let G be a linear operator in the space ℓ 2 defined by the matrix with elements (G) l,k = g(l − k), l, k ≥ 0. The following lemmas give us representations of the functionals P µ , T µ and G.
Let linear operators Ψ µ and Φ in the space ℓ 2 be defined by matrices admits the factorization where b) The linear operator Υ µ in the space ℓ 2 defined by the matrix Proof.Statement a) follows from the equalities Statement b) follows from the equalities Lemma 3 Suppose that functions (33) admit factorizations ( 34) - (36).Let linear operators Ψ µ and Υ µ in the space ℓ 2 be defined as in Lemma 2 and a linear operator Θ µ in the space ℓ 2 be defined by the matrix Then a) Linear operators P µ , T µ and G in the space ℓ 2 admit the factorizations b) An inverse operator V µ = (P µ ) −1 admits the factorization (P µ ) −1 = Θ µ Θ ′ µ and elements of the matrix which determines the operator V µ are calculated by the formula Proof.We give a proof of statement a) for the linear operator P µ only.Factorization (35) implies Thus, For i ≥ j we have the equalities and for i < j we have the equalities that prove statement a).Statement 2) comes from the relation Ψ µ Θ µ = Θ µ Ψ µ = I, which we need to prove.From factorizations (34) and (35) one can obtain These equalities imply the following ones: Lemma 4 Suppose that the function g(λ) admits factorization (36).Let a linear operator S in the space ℓ 2 be defined by a matrix with elements (S) k,j = g(k + j), k, j ≥ 0, and a linear operator K in the space ℓ 2 be defined by a matrix with elements (K) k,j = ϕ(k + j), k, j ≥ 0. Then the operators S and K admit the relation where the linear operator Φ is defined in Lemma 2.
Stat., Optim.Inf.Comput.Vol. 3, June 2015 Proof.In the same way as in the proof of Lemma 3 a) we obtaine the relation Since the matrices S and K are symmetric, we have S = S ′ = Φ ′ K. 2 Under the conditions of Lemma 2 and Lemma 3 on the spectral densities f (λ) and g(λ) formulas ( 22) and ( 23) can be simplified.These lemmas give us the factorizations of the linear functionals T µ and P −1 µ T µ : Denote e µ = GΨ µ a µ .Factorization (35) allows us to make the following transformations: where e µ (m) = (GΨ µ a µ ) m , m ≥ 0, is the mth element of the vector e µ = GΨ µ a µ .Since the following equality holds true: Using factorizations (35) and (36) we make the following transformations: Equalities ( 39) and (40) let us rewrite expression (25) for the spectral characteristic h 2 µ (λ) of the optimal estimate Aη of the functional Aη as where , ψ µ (2), . ..) ′ , C µ is a linear operator defined by a matrix with elements (C µ ) k,j = c µ (k + j), k, j ≥ 0.Here c µ = Sa µ is a vector, S is a linear operator defined by a matrix with elements (S) k,j = g(k + j), k, j ≥ 0. From Lemma 4 the operator S admits representation S = KΦ = Φ ′ K, where K is a linear operator defined by a matrix with elements (K) k,j = ϕ(k + j), k, j ≥ 0.
The spectral characteristic h 1 µ (λ) of the optimal estimate Bξ of the functional Bξ in the case where spectral densities admit canonical factorization ( 34) is of the form where θ µ = (θ µ (0), θ µ (1), θ µ (2), . ..) ′ ; A is a linear operator defined by a matrix with elements (A) k,j = a(k + j), k, j ≥ 0; B µ is a linear operator defined by the matrix with elements This representation of the spectral characteristic h 1 µ (λ) shows that the spectral characteristic h µ (λ) of the estimate Aξ can be calculated by the formula The mean square error of the estimate ∆(f, g; Aξ) is presented as follows: where A µ is a linear operator in the space ℓ 2 defined as for vectors x = (x(0), x(1), x(2), . ..) ′ , y = (y(0), y(1), y(2), . ..) ′ .The obtained results are summarized in the following theorem.
The extrapolation problem for cointegrated stochastic sequences means that we have to find the mean-square optimal linear estimates of the functionals Let the following condition holds true: Then we can determine operators P β µ , T β µ , Q β with the help of the Fourier coefficients of the functions in the same way as we defined operators P µ , T µ , Q in Section 3. It follows from Theorem 4 that the spectral characteristic h β µ (λ) of the optimal estimate of the functional Aξ is calculated by the formula where The mean-square error of the estimate is calculated by the formula Now we can summarize the obtained results in the following statement.

Define operators P
by the Fourier coefficients of functions (47) in the same way as we defined operators P µ , T µ,N , Q N in Section 3. Theorem 5 implies that the spectral characteristic h β µ,N (λ) of the optimal estimate of the functional A N ξ is calculated by the formula where ) k e iλk .The mean-square error of the estimate The following theorem holds true.The spectral characteristic h β µ,N (λ) of the optimal estimate A N ξ is calculated by formula (52).The value of the mean-square error ∆(f, g; A N ξ) is calculated by formula (53).
Suppose that spectral densities f (λ) and p(λ) admit the following canonical factorizations: Define operators K β , Ψ β and Φ β by coefficients of the canonical factorizations (54)-( 56) in the same way as we defined operators K, Ψ and Φ in Section 3. It follows from Theorem 6 that the spectral characteristic h β µ (λ) of the optimal estimate Aξ of the functional Aξ is calculated by the formula where The mean-square error of the estimate is calculated by the formula Theorem 9 Let the cointegrated stochastic sequences {ξ(m), m ∈ Z} and {ζ(m), m ∈ Z} satisfy conditions of Theorem 7.

Minimax-robust method of extrapolation
The values of the mean-square errors ∆(h µ (f, g); f, g) := ∆(f, g; Aξ) and ∆(h µ,N (f, g); f, g) := ∆(f, g; A N ξ) and the spectral characteristics h µ (f, g) and h µ,N (f, g) of the optimal linear estimates Aξ and A N ξ of the functionals Aξ and A N ξ which depend on the unknown values of the sequence ξ(m) based on observations of the stochastic sequence ξ(k) + η(k) can be calculated by formulas (23), (22) and (29), (27) correspondingly under the condition that spectral densities f (λ) and g(λ) of stochastic sequences ξ(m) and η(m) are exactly known.Having canonical factorizations (35) and (36) we can calculate the values of mean-square errors ∆(h µ (f, g); f, g) and spectral characteristics h µ (f, g) by formulas (42), (41) respectively.However, such situation does not appear in practice since we do not know exactly spectral densities of the observed sequences.If in this case we can determine a set D = D f × D g of admissible spectral densities, the minimax (robust) approach to estimation of linear functionals which depend on the unknown values of stochastic sequence with stationary increments can be applied.It consists in finding an estimate that minimizes the maximum value of the mean-square errors for all spectral densities from a given class D = D f × D g of admissible spectral densities simultaneously.
To formalize this approach we present the following definitions.

Definition 5
For a given class of spectral densities D = D f × D g spectral densities f 0 (λ) ∈ D f , g 0 (λ) ∈ D g are called least favorable in the class D for the optimal linear extrapolation of the functional Aξ if the following relation holds true:

Definition 6
For a given class of spectral densities D = D f × D g the spectral characteristic h 0 (λ) of the optimal linear estimate of the functional Aξ is called minimax-robust if there are satisfied the conditions Using the derived in the previous sections formulas and the introduced definitions we can conclude that the following statements hold true.

Lemma 6
Spectral densities f 0 ∈ D f , g 0 ∈ D g which admit canonical factorizations (35) and (36) are least favorable in the class D = D f × D g for the optimal linear extrapolation of the functional Aξ if coefficients {θ 0 (k), ψ 0 (k), ϕ 0 (k) : k ≥ 0} of factorizations (60) determine a solution to the constraint optimisation problem The minimax spectral characteristic is determined as

Lemma 7
Spectral density g 0 ∈ D g which admit canonical factorization (36) with the known spectral density f (λ) is least favorable in the class D g for the optimal linear extrapolation of the functional Aξ if coefficients {θ 0 (k), ψ 0 (k), ϕ 0 (k) : k ≥ 0} of factorizations determine a solution to the constraint optimisation problem The minimax spectral characteristic is determined as If spectral density g(λ) is known and admits canonical factorization (36), extremum problem ( 61) is an extremum problem with respect to variables {θ(k), ψ(k) : k ≥ 0}.

Lemma 8
Spectral density f 0 ∈ D f which admit canonical factorization (35) with the known spectral density g(λ) is least favorable in the class D f for the optimal linear extrapolation of the functional Aξ if coefficients {θ 0 (k), ψ 0 (k) : k ≥ 0} of the factorization determine a solution to the constraint optimisation problem The minimax spectral characteristic is determined as The function h 0 and the pair (f 0 , g 0 ) form a saddle point of the function ∆(h; f, g) on the set H D × D. The saddle point inequalities where or the constraint optimisation problem where Here These constraint optimisation problems (66),(67) are equivalent to the unconstraint optimisation problem [39] ∆ where δ(f, g|D f × D g ) is the indicator function of the set D = D f × D g .Solution (f 0 , g 0 ) to this unconstraint optimisation problem is characterized by the condition 0 ∈ ∂∆ D (f 0 , g 0 ), where ∂∆ D (f 0 , g 0 ) is the subdifferential of the functional ∆ D (f, g) at point (f 0 , g 0 ) ∈ D = D f × D g , that is the set of all continuous linear functionals Λ on L 1 × L 1 which satisfy the inequality ∆ D (f, g) − ∆ D (f 0 , g 0 ) ≥ Λ((f, g) − (f 0 , g 0 )), (f, g) ∈ D (see books [15,39,40] for more details).The form of the functional ∆(h µ (f 0 , g 0 ); f, g) is convenient for application the Lagrange method of indefinite multipliers for finding solution to the problem (68).Making use the method of Lagrange multipliers and the form of subdifferentials of the indicator functions δ(f, g|D f × D g ) of the set D f × D g of spectral densities we describe relations that determine least favourable spectral densities in some special classes of spectral densities (see books [15,33,35] for additional details).

Least favorable spectral densities in the class
Let us assume that densities f 0 ∈ D 0 f , g 0 ∈ D 0 g and the functions are bounded.In this case the functional ∆(h µ (f 0 , g 0 ); f, g) is continuous and bounded in the L 1 × L 1 space.The condition 0 ∈ ∂∆ D (f 0 , g 0 ) leads to the equation for the least favorable densities where α 1 ≥ 0 and α 2 ≥ 0 are such constants that Thus, we have the following statements.

Theorem 10
Let spectral densities f 0 (λ) ∈ D 0 f and g 0 (λ) ∈ D 0 g satisfy condition (15), let functions h µ,f (f 0 , g 0 ) and h µ,g (f 0 , g 0 ) be bounded.The spectral densities f 0 (λ) and g 0 (λ) determined by equations (71), (72) are least favorable in the class D = D 0 f × D 0 g for the optimal linear estimation of the functional Aξ if they determine a solution of extremum problem (59).The function h µ (f 0 , g 0 ) determined by formula (22) is minimax spectral characteristic of the optimal estimate of the functional Aξ.

Theorem 11
Suppose that spectral density f (λ) is known, spectral density g 0 (λ) ∈ D 0 g and conditions (15) is satisfied.Let the function h µ,g (f, g 0 ) be bounded.Spectral density g 0 (λ) is least favorable in the class D 0 g for the optimal linear extrapolation of the functional Aξ if it is of the form } and the pair (f, g 0 ) determines a solution to extremum problem (59).The function h µ (f, g 0 ) determined by formula (22) is minimax spectral characteristic of the optimal estimation of the functional Aξ.
The following statements hold true.

Theorem 17
Let spectral density f (λ) be known, spectral density g 0 (λ) ∈ D ε and condition (15) be satisfied.Assume that the function h µ,g (f, g 0 ) determined by formula (70) is bounded.Spectral density g 0 (λ) is least favorable in the class D ε for the optimal linear extrapolation of the functional Aξ if it is of the form and the pair (f, g 0 ) determines a solution to extremum problem (59).The function h µ (f, g 0 ) determined by formula (22) is minimax spectral characteristic of the optimal estimate of the functional Aξ.

Theorem 18
Suppose spectral density g(λ) is known, spectral density f 0 (λ) ∈ D u v and condition (15) is satisfied.Let the function h µ,f (f 0 , g) be bounded.Spectral density f 0 (λ) is least favorable in the class D u v for the optimal linear extrapolation of the functional Aξ if it is of the form f 0 (λ) = min {v(λ), max {u(λ), g 2 (λ)}} , which depend on the unknown values of the sequence ξ(m) based on observations of the sequence ζ(m) at points m = −1, −2, . ... To find a solution to this problem we can use results presented in the preceding section provided the sequences ξ(m) and ζ(m) − βξ(m) are uncorrelated.

D 0 f × D 0 g
Consider the problem of minimax extrapolation of the functional Aξ which depend on unobserved values of a stochastic sequence ξ(m) with stationary nth increments based on observations of the sequence ξ(m) + η(m) at points of time m = −1, −2, . . .under the condition that spectral densities of the sequences are not known exactly, but the set of admissible spectral densities D = D 0 f × D 0 g is given, where

D 0 g
MINIMAX PREDICTION PROBLEM FOR STOCHASTIC SEQUENCES WITH STATIONARY INCREMENTS 7. Least favorable spectral densities which admit factorization in the class D 0 f × Consider the problem of minimax extrapolation of the functional Aξ from observations ξ(k) + η(k), k ≤ −1, provided that spectral densities f (λ) and g(λ) admit canonical factorizations (35) -(36) and belong to the set D = D 0 f