Robust Extrapolation Problem for Stochastic Processes with Stationary ...

Mathematics and Statistics 2(2): 78-88, 2014 DOI: 10.13189/ms.2014.020204

http://www.hrpub.org

Robust Extrapolation Problem for Stochastic Processes with Stationary Increments Maksym Luz, Mikhail Moklyachuk∗ , Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine ∗ Corresponding

Author: [email protected]

c Copyright ⃝2014 Horizon Research Publishing All rights reserved.

Abstract

The problem of optimal estimation ∫∞ of the linear functionals Aξ = 0 a(t)ξ(t)dt and ∫T AT ξ = 0 a(t)ξ(t)dt depending on the unknown values of stochastic process ξ(t), t ∈ R, with stationary nth increments from observations of the process at points t < 0 is considered. Formulas for calculating the mean square error and the spectral characteristic of optimal linear estimates of the functionals are derived in the case where the spectral density of the process is exactly known. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case where the spectral density of the process is not exactly known, but a set of admissible spectral densities is given.

Keywords

Stochastic process with stationary increments; minimax-robust estimate; mean square error; least favorable spectral density; minimax-robust spectral characteristic

1

Introduction

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 [1], Wiener[2], Yaglom [3, 4]. The further results one can find in book by Rozanov [5]. Yaglom [6, 7] developed a theory of non-stationary processes whose increments of order µ ̸= 0 define a stationary process. The spectral representation for stationary increments and canonical factorization for spectral densities were received, the problem of linear extrapolation of unknown value of stationary stochastic increment from observation of the process was solved. Further results for such stochastic processes were presented by Pinsker [8], Yaglom and Pinsker [9]. See books by Yaglom [3, 4] for more relative results and references. The mean square optimal estimation problems for

stochastic processes with nth stationary increments are natural generalization of the linear extrapolation, interpolation and filtering problems for stationary stochastic processes. The classical methods of extrapolation, interpolation and filtering problems are based on the assumption that the 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 [10] 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 the admissible spectral densities. These estimates are called minimax since they minimize the maximal value of the error. A survey of results in minimax (robust) methods of data processing can be found in the paper by Kassam and Poor [11]. The paper by Grenander [12] should be marked as the first one where the minimax extrapolation problem for stationary processes was formulated and solved. Franke and Poor [13], Franke [14] investigated the minimax extrapolation and filtering problems for stationary sequences with the help of convex optimization methods. This approach makes it possible to find equations that determine the least favorable spectral densities for various classes of admissible densities. In papers by Moklyachuk [18] - [21] 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 details see, for example, books by Kurkin et al.[15], Moklyachuk [16]. Methods of solution the minimax-robust estimation problems for vector stationary sequences and processes were developed by Moklyachuk and Masyutka [17]. Dubovets’ka and Moklyachuk [22] – [28] investigated the minimax-robust estimation problems (extrapolation, interpolation and filtering) for the linear functionals which depend on unknown values of periodically correlated stochastic processes. Luz and Moklyachuk [29, 30] solved the minimax interpolation

Mathematics and Statistics 2(2): 78-88, 2014 ∑N problem for the linear functional AN ξ = k=0 a(k)ξ(k) which depends on unknown (missed) values of a stochastic sequence ξ(m) with stationary nth increments from observations of the sequence with and without noise. This article is dedicated to the mean square optimal estimates of the linear functionals ∫ ∞ ∫ T Aξ = a(t)ξ(t)dt, AT ξ = a(t)ξ(t)dt 0

0

which depend on the unknown values of a stochastic process ξ(t) with stationary nth increments. Estimates are based on observations of the process ξ(t) at points t < 0. The estimation problem for processes with stationary increments is solved in the case of spectral certainty where the spectral density of the process is known as well as in the case of spectral uncertainty where the spectral density of the process is not known, but a set of admissible spectral densities is given. Formulas are derived for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimates of functionals Aξ and AT ξ in the case of spectral certainty where spectral density of the process is known. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case of spectral uncertainty for concrete classes of admissible spectral densities.

2

Stationary stochastic increment process. Spectral representation

Definition 1 For a given stochastic process {ξ(t), t ∈ R} a process ξ (n) (t, τ ) = (1 − Bτ )n ξ(t) =

n ∑ (−1)l Cnl ξ(t − lτ ), (1) l=0

where Bτ is a backward shift operator with step τ ∈ R, such that Bτ ξ(t) = ξ(t − τ ), is called the stochastic nth increment with step τ ∈ R. For the stochastic nth increment process ξ (n) (t, τ ) the following relations hold true: ξ (n) (t, −τ ) = (−1)n ξ (n) (t + nτ, τ ), ξ (n) (t, kτ ) =

∑(k−1)n l=0

Al ξ (n) (t − lτ, τ ),

(2)

∀k ∈ N, (3)

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

(1 + x + . . . + x

) =

exist for all t0 , τ, t, τ1 , τ2 and do not depend on t0 . The function c(n) (τ ) is called the mean value of the nth increment and the function D(n) (t, τ1 , τ2 ) is called the structural function of the stationary nth increment (or the structural function of nth order of the stochastic process {ξ(t), t ∈ R}). The stochastic process {ξ(t), t ∈ R} which determines the stationary nth increment process ξ (n) (t, τ ) by formula (1) is called stochastic process with stationary nth increments. Theorem 1 The mean value c(n) (τ ) and the structural function D(n) (m, τ1 , τ2 ) of a stochastic stationary nth increment process ξ (n) (t, τ ) can be represented in the following forms c(n) (τ ) = cτ n , (4) ∫

D(n) (t, τ1 , τ2 ) = ∞

=

eiλt (1−e−iτ1 λ )n (1−eiτ2 λ )n

−∞

(1 + λ2 )n dF (λ), (5) λ2n

where c is a constant, F (λ) is a left-continuous nondecreasing bounded function with F (−∞) = 0. The constant c and the function F (λ) are determined uniquely by the increment process ξ (n) (t, τ ). From the other hand, a function c(n) (τ ) which has the form (4) with a constant c and a function D(n) (m, τ1 , τ2 ) which has the form (5) with a function F (λ) which satisfies the indicated conditions are the mean value and the structural function of some stochastic stationary nth increment process ξ (n) (t, τ ). Using representation (5) of the structural function of the stationary nth increment process ξ (n) (t, τ ) and the Karhunen theorem (see Karhunen [31]), we get the following spectral representation of the stationary nth increment process ξ (n) (t, τ ): ∫ ∞ (1 + iλ)n (n) ξ (t, τ ) = eitλ (1 − e−iλτ )n dZ(λ), (6) (iλ)n −∞ where Z(λ) is an orthogonal stochastic measure on R connected with the spectral function F (λ) by the relation EZ(A1 )Z(A2 ) = F (A1 ∩ A2 ) < ∞.

(7)

Denote by H(ξ (n) ) the subspace of the Hilbert space H = L2 (Ω, F, P) of the second order stochastic variables which is generated by elements {ξ (n) (t, τ ) : t, τ ∈ R} and let H t (ξ (n) ), t ∈ R, be a subspace of H(ξ (n) ) generated by elements {ξ (n) (u, τ ) : u ≤ t, τ > 0}. Let S(ξ (n) ) be defined by the relationship ∩ S(ξ (n) ) = H t (ξ (n) ). t∈R

(k−1)n k−1 n

79

l

Al x .

l=0

Definition 2 The stochastic nth increment process ξ (n) (t, τ ) generated by stochastic process {ξ(t), t ∈ R} is wide sense stationary if the mathematical expectations Eξ (n) (t0 , τ ) = c(n) (τ ), Eξ (n) (t0 + t, τ1 )ξ (n) (t0 , τ2 ) = D(n) (t, τ1 , τ2 )

Since the space S(ξ (n) ) is a subspace of the Hilbert space H(ξ (n) ), the space H(ξ (n) ) admits the decomposition H(ξ (n) ) = S(ξ (n) ) ⊕ R(ξ (n) ), where R(ξ (n) ) is an orthogonal complement of the subspace S(ξ (n) ) in the space H(ξ (n) ). In the following we will consider increments ξ (n) (t, τ ) with a step τ > 0.

80

Robust Extrapolation Problem for Stochastic Processes with Stationary Increments

Definition 3 A stationary nth increment process ξ (n) (t, τ ) is called regular if H(ξ (n) ) = R(ξ (n) ). It is called singular if H(ξ (n) ) = S(ξ (n) ). Theorem 2 A wide-sense stationary stochastic increment process ξ (n) (t, τ ) admits a unique representation in the form ξ (n) (t, τ ) = ξr(n) (t, τ ) + ξs(n) (t, τ ),

(8)

(n)

where {ξr (t, τ ) : t ∈ R} is a regular increment process (n) and {ξs (t, τ ) : t ∈ R} is a singular increment pro(n) cess. Moreover, the increment processes ξr (z, τ ) and (n) ξs (t, τ ) are orthogonal for all t, z ∈ R. Components of the representation (8) are constructed in the following way: ξs(n) (t, τ ) = E[ξ (n) (t, τ )|S(ξ (n) )], ξr(n) (t, τ ) = ξ (n) (t, τ ) − ξs(n) (t, τ ). Let {η(t) : t ∈ R} be a stochastic process with independent increments such that E|η(t) − η(s)|2 = |t − s| and H z (ξ (n) ) = H z (η) for all z ∈ R, where the subspace H z (η) of the space H is generated by values {η(u) : u ≤ z} of the process η(t). Defined stochastic process is called innovation process. Theorem 3 A stochastic stationary increment process ξ (n) (t, τ ) is regular if and only if there exists an innovation process : t ∈ R} and a function {φ(n) (t, τ ) : ∫ ∞ {η(t) (n) t ≥ 0}, 0 |φ (t, τ )|2 dt < ∞, such that ∫ ∞ ξ (n) (t, τ ) = φ(n) (u, τ )dηu (t − u), (9)

Φτ (λ) =

(1 − e−iλτ )n (1 + iλ)n Φ(λ). (iλ)n

The one-sided moving average representation (9) is used for finding the optimal mean square estimate of the unknown values of a process ξ(t) based on observations of the process at points t < 0.

3

Hilbert space projection method of extrapolation of linear functionals

Let a stochastic process {ξ(t), t ∈ R} defines nth increment process ξ (n) (t, τ ) with an absolutely continuous spectral function F (λ) which has spectral density f (λ). Without loss of generality we will assume that the mean value of the increment process ξ (n) (t, τ ) equals to 0. Let the stationary increment process ξ (n) (t, τ ) admit the one-sided moving average representation (9) and the spectral density f (λ) admits the canonical factorization (11). Consider the case where the step τ > 0. Let the values of the process ξ(t) be known for t < 0. The problem is to find the mean square optimal lin∫T ear estimates of functionals AT ξ = 0 a(t)ξ(t)dt and ∫∞ Aξ = 0 a(t)ξ(t)dt which depend on unknown values ξ(t), t ≥ 0. In order to solve the stated problem we will present the process ξ(t), t ≥ 0, as a sum of its increments ξ (n) (t, τ ), t ≥ 0, τ > 0, and its initial values ξ 0 . Particularly, when τ ∗ > t∗ ≥ 0, a relation ξ(t∗ ) = ξ (n) (t∗ , τ ∗ ) +

0

0

where 0 |φ (t, τ )| dt < ∞ and η(t), t ∈ R, is an innovation process. (n)

2

Let the stationary nth increment process ξ (n) (t, τ ) admit the canonical representation (9). In this case the spectral function F (λ) of the stationary increment process ξ (n) (t, τ ) has spectral density f (λ) which admits the canonical factorization ∫ ∞ f (λ) = |Φ(λ)|2 , Φ(λ) = e−iλt φ(t)dt. (11)

comes from (1), where ξ 0 = {ξ(t∗ − lτ ∗ ) : l = 1, 2, . . . , n} ⊂ {ξ(t) : t ≤ 0} are known observations. The following lema describes a representation of the functional Aξ that depends on known initial values of the process ξ(t) and its increments ξ (n) (t, τ ) for t ≥ 0 in the case of arbitrary step τ > 0. ∫∞ Lema 1 A linear functional Aξ = 0 a(t)ξ(t)dt admits a representation Aξ = Bξ − V ξ, where ∫ ∞ ∫ 0 Bξ = bτ (t)ξ (n) (t, τ )dt, V ξ = vτ (t)ξ(t)dt, −τ n

0

vτ (t) =

n ∑ l=[− τt ]

0

Let us define ∫ Φτ (λ) =

∞

e−iλt φ(n) (t, τ )dt =

0

∫

bτ (t) = ∞

e−iλt φτ (t)dt,

0

where φτ (t) = φ(n) (t, τ ) is the function from the representation (9). Defined function Φτ (λ), which is a Fourier transform of the function φ(n) (t, τ ), is related with spectral density f (λ) of the stochastic process ξ (n) (t, τ ) by relations 2

|Φτ (λ)| =

|1 − e

n ∑ (−1)l Cnl ξ(t∗ − lτ ∗ ) l=1

Conclusion 1 Using theorems 2 and 3 one can conclude that a wide-sense stationary stochastic increment process admits a unique representation in the form ∫ ∞ ξ (n) (t, τ ) = ξs(n) (t, τ ) + φ(n) (u, τ )dηu (t − u), (10) ∫∞

(13)

−iλτ 2n

| (1 + λ ) f (λ), λ2n

∞ ∑

(−1)l Cnl bτ (t + lτ ),

a(t + τ k)d(k) = Dτ a(t), t ≥ 0,

(12)

(15)

k=0

where [x]′ denotes the least integer number among numbers that are greater or equal to x, {d(k) : k ≥ 0} are ∑∞ coefficients determined by the relation d(k)xk = k=0 (∑ )n ∞ j , Dτ is a linear transformation which acts j=0 x on an arbitrary function x(t), t > 0, by formula Dτ x(t) =

2 n

t ∈ [−τ n; 0), (14)

′

∞ ∑ k=0

x(t + τ k)d(k).

(16)

Mathematics and Statistics 2(2): 78-88, 2014 b 2 = ∆(f, Bξ). b = E|Bξ − Bξ|

Proof. From (1) we can obtain the formal equation ∞

∑ 1 (n) ξ(t) = ξ (t, τ ) = d(j)ξ (n) (t − τ j, τ ), (1 − Bτ )n j=0 (17) and the relations ∫ ∞ ∫ a(t)ξ(t)dt = − 0

∫

(

∞

+ 0

vτ (t)ξ(t)dt+ )

∞ ∑

a(t + τ k)d(k) ξ (n) (t, τ )dt,

k=0

∫

∞

bτ (k)ξ

(n)

where hτ (λ) is the spectral characteristic of the estimate b The spectral characteristic of the optimal estimate Bξ. provides the minimum value of the mean square error b ∆(f, Bξ). Let the stochastic increment ξ (n) (t, τ ) admits the canonical representation (9). Then the functional Bξ can be presented by formulas ∫ ∞∫ ∞ Bξ = bτ (t)φ(u, τ )dηu (t − u)dt =

(t, τ )dt =

0

∫

n ∑

0

ξ(t)

= −τ n

∫

l=[− τt ]

∞

+

ξ(t) 0

Denote by L0− 2 (f ) the subspace of the Hilbert space L2 (f ) generated by the set of functions ∫ (1 + iλ)n ∞ h(λ) = (1 − e−iλτ )n h(t)e−iλt dt. (iλ)n 0 b of the functional Bξ admits a Every linear estimate Bξ representation ∫ ∞ (1 + iλ)n b = Bξ hτ (λ)(1 − e−iλτ )n dZ(λ), (21) (iλ)n −∞

0 −τ n

(−1)l Cnk bτ (t + τ l)dt+ ′

n ∑ (−1)l Cnk bτ (t + τ l)dt. l=0

0

∫

From two last relations we can get the representation of the functional Aξ and relations (15), (14).

∫

∫

T

bτ,T (t)ξ (n) (t, τ )dt, VT ξ = 0

0 −τ n

vτ,T (t)ξ(t)dt,

We will suppose that the following restrictions on the function bτ (t) hold true ∫ ∞ ∫ ∞ |bτ (t)|dt < ∞, t|bτ (t)|2 dt < ∞. (18) 0

Under these conditions the functional Bξ has the second moment and the operator Bτ defined below is compact. Since the functions a(t) and bτ (t) are related by (15), the following conditions hold true ∫ ∞ ∫ ∞ τ |D a(t)|dt < ∞, t|Dτ a(t)|2 dt < ∞. (19) 0

0

b denote the mean square optimal linear esLet Aξ timate of the functional Aξ from observations of the b denote the mean process ξ(t) for t < 0 and let Bξ square optimal linear estimate of the functional Bξ from observations of the stochastic nth increment process b := E|Aξ − Aξ| b 2 deξ (n) (t, τ ) for t < 0. Let ∆(f, Aξ) b and let note the mean square error of the estimate Aξ b := E|Bξ − Bξ| b 2 denote the mean square error ∆(f, Bξ) b Since values ξ(t) for t ∈ [−τ n; 0) are of the estimate Bξ. known, the following equality comes from lema 1: b = Bξ b − V ξ. Aξ Thus b = E|Aξ − Aξ| b 2 = E|Aξ + V ξ − Bξ| b 2= ∆(f, Aξ)

(20)

∞

−∞ 0 ∫ ∞∫ ∞

bτ (t)φ(t − u, τ )dtdη(u)+ bτ (t)φ(t − u, τ )dtdη(u).

+ 0

and functions bτ,T (t), t ∈ [0, T ], and vτ,T (t), t ∈ [−τ n; 0), are defined by formulas (14) and (15) respectively under the condition a(t) = 0 for t > T .

0

0

∫

0

=

Conclusion 2 The linear functional AT ξ admits a representation AT ξ = BT ξ − VT ξ, where BT ξ =

81

u

As the relation H 0 (ξ (n) ) = H 0 (η) holds true and increments of the process η(t) are orthogonal, the optimal b of the functional Bξ is calculated as estimate Bξ ∫ 0 ∫ ∞ b Bξ = bτ (t)φτ (t − u)dtdη(u) = −∞

∫

0

∫

0

+ −∞

Bφτ (t − u)dη(u) =

∞

B ∗ φτ (λ)dη ∗ (λ), (22)

−∞

where B ∗ φτ (λ) and η ∗ (λ) are ∫ ∞inverse Fourier transforms of the function Bφτ (u) = 0 bτ (t)φτ (t − u)dt, u < 0, and the process η(u) respectively. Yaglom[6] showed that ∫ λ dZ(p) η ∗ (λ) = . (23) −∞ Φ(p) So we need to find B ∗ φτ (λ). ∫ ∞ ∫ 0 bτ (t)φτ (t − s)dtds = B ∗ φτ (λ) = eiλs ∫

−∞

∞

= 0

0

∫

∞

eiλt bτ (t) 0

∫

∞

=

iλt

e 0

e−iλ(s+t) φτ (t + s)dsdt =

∫

∞

bτ (t)

∫

= Bτ (λ)Φτ (λ) −

∞

0 ∞

∫ = Bτ (λ)Φτ (λ)− 0

e−iλz φτ (z)dzdt =

t

∫

eiλt bτ (t) ∫

eiλy

t

e−iλz φτ (z)dzdt =

0 ∞

bτ (y+z)φτ (z)dzdy. (24) 0

Substituting the expressions (23) and (24) in (22) and using (13) one can obtain the following formulas for culculating the spectral characteristic of the optimal estib mate Bξ: hτ (λ) = B τ (λ) − rτ (λ)Φ−1 τ (λ),

(25)

82

Robust Extrapolation Problem for Stochastic Processes with Stationary Increments ∫

∞

B τ (λ) =

∫ bτ (t)eiλt dt, rτ (λ) =

0

∞

eiλt (Bτ φτ )(t)dt,

0

where Bτ is a linear operator in L2 ([0, ∞)) space which defined by the relation ∫ ∞ (Bτ φτ )(t) = bτ (t + u)φτ (u)du. 0

Making use relation (20) we can find the optimal esb of the functional Aξ from observations of the timate Aξ process ξ(t) for t < 0. These estimate can be presented in the following form: ∫ 0 b Aξ = − vτ (t)ξ(t)dt+ ∫

(n)

Here φτ (u) = φ (u, τ ) is the function from the moving average representation (9). The operator Bτ is compact providing (18). b can be The value of the mean square error ∆(f, Bξ) calculated by the formula b = E|Bξ − Bξ| b 2= ∆(f, Bξ) ∫ ∞ 1 = |rτ (λ)|2 dλ = ||Bτ φτ ||2 . 2π −∞

(26)

Theorem 4 Let a stochastic process {ξ(t), m ∈ Z} determine a stationary stochastic nth increment process ξ (n) (t, τ ) with absolutely continuous spectral function F (λ) and spectral density f (λ) which admits the canonb ical factorization (11). The optimal linear estimate Bξ of the functional Bξ which depends on the unobserved values {ξ (n) (t, τ ) : t ≥ 0}, τ > 0, from observations of the process ξ(t) for t < 0 can be calculated by formula (21). The spectral characteristic hµ (λ) of the optimal b can be calculated by formula (25). linear estimate Bξ b can be calThe value of the mean square error ∆(f, Bξ) culated by formula (26). Using Theorem 4 and representation (8), we can obtain the optimal estimate of an unobserved value ξ (n) (u, τ ), τ > 0, at the point u ≥ 0 from observations of the process ξ(t) for t < 0. The singular component (n) ξs (u, τ ) of representation (8) of the process has errorless estimate. We will use formula (25) to obtain the spectral characteristic hu,τ (λ) of the optimal estimate (n) ξb(n) (t, τ ) of the regular component ξr (u, τ ) of the process. Consider a function B τ (λ) = eiλu . It follows from the derived formulas that the spectral characteristic of the estimate ∫

b(n)

ξ

(u, τ ) =

ξs(n) (u, τ )+

∞

(1 + iλ)n hu,τ (λ)(1 − e−iλτ )n + dZ(λ) (iλ)n −∞ can be calculated by the formula ∫ u hu,τ (λ) = eiλu − Φ−1 (λ) φτ (y)e−iλy dy. τ

(27)

(28)

0

The value of the mean square error can be calculated by the formula ∫ u 1 |φτ (y)|2 dy. (29) ∆(f, ξb(n) (u, τ )) = 2π 0 The following statement holds true. Conclusion 3 The optimal linear estimate ξb(n) (u, τ ) of the value ξ (n) (u, τ ), τ > 0, at the point u ≥ 0 of the increment process ξ (n) (t, τ ) from observations of the process ξ(t), t < 0, can be calculated by formula (27). The spectral characteristic hu,τ (λ) of the optimal linear estimate ξb(n) (u, τ ) can be calculated by formula (28). The value of mean square error ∆(f, ξb(n) (u, τ )) of the optimal linear estimate can be calculated by formula (29).

−τ n

∞

+ −∞

−iλτ n h(a) ) τ (λ)(1 − e

(1 + iλ)n dZ(λ), (iλ)n

(30)

where the function vτ (t), t = [−τ n, 0), is defined by relation (14). Using the relationship (15) between the functions a(t) and bτ (t), t ≥ 0, we obtain the following equation: ∫ ∞ τ (B φτ )(t) = Dτ a(t + u)φ(u, τ )du = ∫ = Dτ

0 ∞

a(t + u)φ(u, τ )du = Dτ (Aφτ )(t),

0

where the linear operator A is defined by the function a(t), t ≥ 0, in the following way: ∫ ∞ (Aφτ )(t) = a(t + u)φτ (u)du. 0

Thus the spectral characteristic and the value of the b can be mean square error of the optimal estimate Aξ calculated by the formulas −1 (a) h(a) τ (λ) = Aτ (λ) − rτ (λ)Φτ (λ), ∫ ∞ Aτ (λ) = Dτ a(t)eiλt dt, 0 ∞

∫ rτ(a) (λ) =

Dτ (Aφτ )(t)eiλt dt.

(31)

(32)

0

b = E|Aξ − Aξ| b 2= ∆(f, Aξ) ∫ ∞ 1 = |r(a) (λ)|2 dλ = ||Dτ Aφτ ||2 . 2π −∞ τ

(33)

The following theorem holds true. Theorem 5 Let a stochastic process {ξ(t), t ∈ R} determine a stationary stochastic nth increment process ξ (n) (t, τ ) with absolutely continuous spectral function F (λ) and spectral density f (λ) which admits the canonb ical factorization (11). The optimal linear estimate Aξ of the functional Aξ of unobserved values ξ(t), t ≥ 0, from observations of the process ξ(t) for t < 0 can be calculated by formula (30). The spectral characteristic (a) b can be calcuhµ (λ) of the optimal linear estimate Aξ lated by formula (31). The value of the mean square b of the optimal linear estimate can be calerror ∆(f, Aξ) culated by formula (33). Consider now the problem of the mean square optimal estimation of the functional AT ξ. The optimal estimate of the functional can be calculated by formula ∫ 0 bT ξ = − A vτ,T (t)ξ(t)dt+ ∫

−τ n

∞

+ −∞

hτ,T (λ)(1 − e−iλτ )n (a)

(1 + iλ)n dZ(λ), (iλ)n

(34)

Mathematics and Statistics 2(2): 78-88, 2014

where the function vτ,T (t), t ∈ [−τ n; 0), can be calculated by formulas

point u ≥ 0 from observations ξ(t) for t < 0 can be calculated by formula

min{[ T τ−t ],n}

∑

vτ,T (t) =

(−1)

l

Cnl bτ,T (lτ

b = ξ(u)

+ t), t ∈ [−τ n; 0),

bτ,T (t) =

∫

]

T −t [∑ τ ]

−∞

k=0

Here DτT is a linear transformation which acts on an arbitrary function x(t), t ∈ [0, T ], as DτT x(t)

=

T −t [∑ τ ]

x(t + τ k)d(k).

k=0 (a)

The spectral characteristic hτ,T (λ) and the value of bT ξ) of the estimate A bT ξ the mean square error ∆(f, A can be calculated by formulas hτ,T (λ) = Aτ,T (λ) − rτ,T (λ)Φ−1 τ (λ), (a)

(a)

∫

(35)

T

DτT a(t)eiλt dt,

Aτ,T (λ) = 0

∫ (a) rτ,T (λ)

T

DτT (AT φτ )(t)eiλt dt,

=

(36)

0

where AT is a linear operator in L2 ([0, ∞)) space defined by formula T −t

(AT φτ )(t) =

=

a(t + u)φτ (u)du, DτT AT φτ

T −t [∑ τ ]∫

k=0

(1 + iλ)n dZ(λ). (iλ)n

(38)

The spectral characteristic hu,τ (λ) and the value of the b mean square error ∆(f, ξ(u)) = ∆(f, ξb(n) (u, τ )) of the b estimate ξ(u) can be calculated by formulas (28) and (29) respectively. Consequently, the following statement holds true. Conclusion 4 Let τ > u ≥ 0. The optimal mean b square estimate ξ(u) of the unknown value of the element ξ(u) from observations of the process ξ(t) for t < 0 can be calculated by formula (38). The spectral characb teristic hu,τ (λ) of the optimal linear estimate ξ(u) can be calculated by formula (28). The value of mean square b error ∆(f, ξ(u)) can be calculated by formula (29). Remark 1 Using relation (12) we can find a relationship between functions φτ (t), t ≥ 0, and φ(t), t ≥ 0. Since ∫ ∞ |1 − e−iλτ |2n (1 + λ2 )n 1 ln 1 + λ2 dλ < ∞ λ2n −∞ for all n ≥ 1 and τ > 0, there are functions ωτ (t), t ≥ 0, and ∫ ∞ ωτ (λ)e−iλt dt

Ωτ (λ) = such that

0

and linear opperator defined by formula

hu,τ (λ)(1 − e−iλτ )n

0

∫

DτT (AT φτ )(t)

∞

+ a(t + τ k)d(k) = DτT a(t), t ∈ [0; T ].

T ∑ (−1)l+1 Cnl ξ(u − lτ )+ l=1

′ − τt

l=[

83

1 ||ωτ (λ)|| = 2π

∫

∞

2

in L2 ([0, ∞)) space is

T −t−τ k

|ωτ (λ)|2 dt < ∞,

0

|1 − e−iλτ |2n (1 + λ2 )n = |Ωτ (λ)|2 λ2n and the following relation holds true:

a(u+t+τ k)φτ (u)d(k)du; Φτ (λ) = Ωτ (λ)Φ(λ).

0

bT ξ) = E|AT ξ − A bT ξ|2 = ∆(f, A ∫ ∞ 1 (a) = |r (λ)|2 dλ = ||DτT AT φτ ||2 . 2π −∞ τ,T

(37)

Consequently, the following theorem holds true.

From (39) using the inverse Fourier transform we get ∫ ∞ eiλt Ωτ (λ)Φ(λ)dλ = φτ (t) = ∫ =

−∞

∫

∞

Consider the case where τ > u ≥ 0. In this case the optimal mean square estimate of the value ξ(u) at the

∞

φ(x) 0

Theorem 6 Let a stochastic process {ξ(t), t ∈ R} determine a stationary stochastic nth increment process ξ (n) (t, τ ) with absolutely continuous spectral function F (λ) and spectral density f (λ) which admits the canonibT ξ cal factorization (11). The optimal linear estimate A of the functional AT ξ of unobserved values ξ(t), t ≥ 0, from observations of the process ξ(t) for t < 0 can be calculated by formula (34). The spectral characteristic (a) bT ξ can be calcuhτ,T (λ) of the optimal linear estimate A lated by formula (35). The value of mean square error bT ξ) can be calculated by formula (37). ∆(f, A

(39)

∫

−∞

eiλ(t−x) Ωτ (λ)dλdx =

t

ωτ (t − x)φ(x)dx.

= 0

Therefore, the functions φτ (t), t ≥ 0, and φ(t), t ≥ 0, from the space L2 ([0, ∞)) are related by the relation ∫ t φτ (t) = Wτ φ(t) = ωτ (t − x)φ(x)dx, (40) 0

where Wτ is a linear operator in the space L2 ([0, ∞)) defined by the function ωτ (t), t ≥ 0, from the space L2 ([0, ∞)). In the case where the functions φτ (t) and φ(t) are defined on segment [0, T ], which means φτ (t) = φ(t) = 0 for t > T , the relation between functions is defined by (40) for t ∈ [0, T ].

84

Robust Extrapolation Problem for Stochastic Processes with Stationary Increments

4

Minimax-robust method of extrapolation

The proposed formulas may be employed under the condition that the spectral density f (λ) of the considered stochastic process ξ(t) with stationary nth increments is known. The value of the mean square error (a) b and the spectral characteris∆(hτ (f ); f ) := ∆(f, Aξ) (a) b of the functic hτ (f ) of the optimal linear estimate Aξ tional Aξ which depends of unknown values ξ(t) can be calculated by formulas (31) and (33), the value of (a) bT ξ) and the mean square error ∆(hτ,T (f ); f ) := ∆(f, A (a)

spectral characteristic hτ,T (f ) of the optimal linear esbT ξ of the functional Aξ which depends of untimate A known values ξ(t), t ≥ 0, can be calculated by formulas (35) and (37). In the case where the spectral density is not known, but a set D of admissible spectral densities is given, the minimax (robust) approach to estimation of the functionals of the unknown values of a stochastic process with stationary increments is reasonable. In other words we are interesting in finding an estimate that minimizes the maximum of the mean square errors for all spectral densities from a given class D of admissible spectral densities simultaneously.

Lema 3 Spectral density f 0 (λ) ∈ D which admits the canonical factorization (11) is the least favorable in the class of admissible spectral densities D for the optimal linear estimation of the functional AT ξ if ∫ 2 T 0 0 −iλt f (λ) = φ (t)e dt , (43) 0 where φ0 (t), t ∈ [0; T ], is a solution to the conditional extremum problem ∫ 2 T ||DτT AT φτ ||2 → max, f (λ) = φ(t)e−iλt dt ∈ D. 0 (44) (a)

If hτ (f 0 ) ∈ HD , the minimax-robust spectral char(a) acteristic can be calculated as h0 = hτ (f 0 ). The minimax-robust spectral characteristic h0 and the least favorable spectral density f 0 form a saddle point of the function ∆(h; f ) on the set HD × D. The saddle point inequalities ∆(h; f 0 ) ≥ ∆(h0 ; f 0 ) ≥ ∆(h0 ; f ) ∀f ∈ D, ∀h ∈ HD (a)

Definition 4 For a given class of spectral densities D a spectral density f0 (λ) ∈ D is called least favorable in D for the optimal linear estimate the functional Aξ if the following relation holds true: 0 0 (a) ∆(f 0 ) = ∆(h(a) τ (f ); f ) = max ∆(hτ (f ); f ). f ∈D

Definition 5 For a given class of spectral densities D a 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 =

L0− 2 (f ),

f ∈D

min max ∆(h; f ) = sup ∆(h0 ; f ).

h∈HD f ∈D

f ∈D

Analyzing the derived formulas and using the introduced definitions we can conclude that the following statements are true. Lema 2 Spectral density f (λ) ∈ D which admits the canonical factorization (11) is the least favorable in the class of admissible spectral densities D for the optimal linear estimation of the functional Aξ if 0

∫ f (λ) =

∞

0

0

φ (t)e

−iλt

0

2 dt ,

(41)

where φ0 (t), t ∈ [0; ∞) is a solution to the conditional extremum problem ||D Aφτ || → max, τ

2

∫ f (λ) =

0

∞

−iλt

φ(t)e

2 dt ∈ D. (42)

(a)

hold true if h0 = hτ (f 0 ) and hτ (f 0 ) ∈ HD , where f 0 is a solution to the conditional extremum problem 0 e ) = −∆(h(a) ∆(f f ∈ D, τ (f ); f ) → inf, ∫ ∞ 1 |rτ (λ)|2 0 ∆(h(a) f (λ)dλ. τ (f ); f ) = 2π −∞ f 0 (λ)

(45)

Here rτ is determined by formula (32) or (36) with f (λ) = f 0 (λ). The conditional extremum problem (45) is equivalent to the unconditional extremum problem e ) + δ(f |D) → inf, ∆D (f ) = ∆(f where δ(f |D) is the indicator function of the set D. Solution f 0 to this unconditional extremum problem is characterized by the condition 0 ∈ ∂∆D (f 0 ) (see Pshenichnyi[32]), where ∂∆D (f 0 ) is the subdifferential of the functional ∆D (f 0 ) at point f 0 . With the help of the condition 0 ∈ ∂∆D (f 0 ) we can find the least favorable spectral densities in some special classes of spectral densities (see books by Moklyachuk[16], Moklyachuk and Masyutka[17] for more details).

5

Least favorable spectral densities in the class D0

Consider the problem of the optimal estimation of functionals Aξ and AT ξ of unknown values ξ(t), t ≥ 0, of the stochastic process ξ(t) with stationary nth increments in the case where the spectral density is not known, but the following set of spectral densities is given } { ∫ ∞ 1 f (λ)dλ ≤ P0 . D0 = f (λ)| 2π −∞ It follows from the condition 0 ∈ ∂∆D (f 0 ) for D = D0 that the least favorable density satisfies the equation |rτ(a) (λ)|2 (f 0 (λ))−1 = ψ(λ) + c−2 ,

Mathematics and Statistics 2(2): 78-88, 2014

where ψ(λ) ≤ 0 and ψ(λ) = 0 if f 0 (λ) > 0. Therefore, the least favorable density in the class D0 for the optimal linear estimation of the functional Aξ can be presented in the form ∫ ∞ 2 0 τ 0 iλt f (λ) = c D (Aφτ )(t)e dt , (46)

Taking into consideration (40), we have the following equality 2 2 ∫ T (a) DτT (AT Wτ φ)(t)eiλt dt = rτ,T (λ) = 0 ∫ 2 T τ b τ −iλt = DT (AT W φ)(t)e dt , 0

0

where the unknown function cφ0τ (t) can be calculated using factorization ∫ ∞ (11), equation (40), condition (42) and condition −∞ |φ0 (λ)|2 dλ = 2πP0 . Consider the equation Dτ AWτ φ = αφ,

α ∈ C.

(47)

b T φτ )(t) DτT (A

=

[ τt ] ∫ ∑

∞

τ

τ

D (AW φ)(t)e 0

iλt

2 dt .

Denote by ν0 P0 the maximum value of ||Dτ AWτ φ||2 on the set of those solutions φ of equation (47), which satisfy condition ||φ||2 = P0 and define canonical factorization (11) of the spectral density f 0 (λ). Let ν0+ P0 be the maximum value of ||Dτ AWτ φ||2 on the set of those φ which satisfy condition ||φ||2 = P0 and define canonical factorization (11) of the spectral density f 0 (λ) defined by (46). The derived equations and conditions give us a possibility to verify the validity of following statement. Theorem 7 If there exists a solution φ0 = φ0 (t) of equation (47) which satisfies conditions ||φ0 ||2 = P0 and ν0 P0 = ν0+ P0 = ||Dτ AWτ φ0 ||2 , the spectral density (41) is the least favorable density in the class D0 for the optimal estimation of the functional Aξ of unknown values ξ(t), t ≥ 0, of the stochastic process ξ(t) with stationary nth increments. The increment ξ (n) (t, τ ) admits a one-sided moving average representation. If ν0 < ν0+ , the density (46) which admits the canonical factorization (11) is the least favorable in the class D0 . The function cφτ = cφτ (t) is determined by equality (40), ∫∞ condition (42) and condition −∞ |φ(λ)|2 dλ = 2πP0 . The minimax-robust spectral characteristic is calculated by formulas (31), (32) substituting f (λ) by f 0 (λ). Consider the problem of optimal estimation of the functional AT ξ. In this case the least favorable spectral density is determined by the relation ∫ 2 T τ 0 iλt f (λ) = c DT (AT φτ )(t)e dt . 0 0

b T in the space L2 ([0, ∞)) by Define a linear operator A relation ∫ t b T φτ )(t) = (A a(T − t + u)φτ (u)du. (49) 0

a(T −t+u+τ k)φτ (u)d(k)du.

0

or the equation b T Wτ φ = βφ, DτT A

β ∈ C,

(52)

such that ||φ||2 = P0 , satisfies the following equality ∫ 2 N 2 (a) 0 −iλt f (λ) = φ(t)e dt = crτ,T (λ) . 0 Denote by ν0T P0 the maximum value of b T Wτ φ||2 on the set of ||DτT AT Wτ φ||2 = ||DτT A solutions φT of the equation (51) or the equation (52), which satisfy condition ||φ||2 = P0 and determine the canonical factorization (11) of the spectral density f 0 (λ) ∈ D0 . Let ν0N + P0 be the maximum value of ||DτT AT Wτ φ||2 on the set of those φ which satisfy condition ||φ||2 = P0 and determine the canonical factorization (11) of the spectral density f 0 (λ) defined by (48). The following statement holds true. Theorem 8 If there exists a solution φ0 = φ0 (t), t ∈ [0; T ] of equation (51) or equation (52) such that ||φ0 ||2 = P0 and ν0T P0 = ν0T + P0 = ||DτT AT Wτ φ0 ||2 , the spectral density (43) is least favorable in the class D0 for the optimal estimation of the functional AT ξ of unknown values ξ(t), t ∈ [0; T ], of the stochastic process ξ(t) with stationary nth increments. The increment ξ (n) (t, τ ) admits a one-sided moving average representation. If ν0T < ν0T + , the density (48) which admits the canonical factorization (11) is the least favorable in the class D0 . The function cφτ = cφτ (t), t ∈ [0; T ], is determined by equation (40), condition (44) and condi∫∞ tion −∞ |φ(λ)|2 dλ = 2πP0 . The minimax-robust spectral characteristic is calculated by formulas (35), (36) substituting f (λ) by f 0 (λ).

6 (48)

t−τ k

Therefore each solution φ = φ(t), t ∈ [0, T ], of the equation (51) DτT AT Wτ φ = αφ, α ∈ C,

0

∫ = c

(50)

b T φτ in the space where the linear operator DτT A L2 ([0, ∞)) is calculated by formula

k=0

For each solution of this equation such that ||φ||2 = 1 ∫∞ |φ(λ)|2 dλ = P0 the following relation holds true: 2π −∞ ∫ ∞ 2 0 −iλt f (λ) = φ(t)e dt =

85

Least favorable spectral densities in the class Dvu

Consider the case where the spectral density f (λ) is not known, but the following set of spectral densities is given { } ∫ ∞ u Dv = f (λ)|v(λ) ≤ f (λ) ≤ u(λ), f (λ)dλ = 2πP0 , −∞

86

Robust Extrapolation Problem for Stochastic Processes with Stationary Increments

where v(λ) and u(λ) are some given (fixed) spectral densities. It follows from the condition 0 ∈ ∂∆D (f 0 ) for D = Dvu that the least favorable density f 0 (λ) in the class Dvu for the optimal linear estimation of the functional Aξ is of the form { { 2 }} f 0 (λ) = max v(λ), min u(λ), s0τ (λ) , (53) ∫ ∞ s0τ (λ) = c Dτ (Aφ0τ )(t)eiλt dt, 0

where the unknown function cφ0τ (t) can be calculated using∫ factorization (11), equestion (40), conditions (42) ∞ and −∞ |φ0 (λ)|2 dλ = 2πP0 . Denote by νuv P0 the maximum value of ||Dτ AWτ φ||2 on the set of those solutions φ of equation (47) which satisfy condition ||φ||2 = P0 , inequalities ∫ ∞ 2 v(λ) ≤ φ(t)e−iλt dt ≤ u(λ) 0

and determine the canonical factorization (11) of the + spectral density f (λ). Let νuv P0 be the maximum value τ τ 2 of ||D AW φ|| on the set of those φ which satisfy condition ||φ||2 = P0 and determine the canonical factorization (11) of the spectral density f 0 (λ) defined by (53). The derived equations and conditions give us a possibility to verify the validity of the following statement. Theorem 9 If there exists a solution φ0 = φ0 (t) of equation (47) which satisfies conditions ||φ0 ||2 = P0 and + νuv P0 = νuv P0 = ||Dτ AWτ φ0 ||2 , the spectral density (41) is the least favorable in the class Dvu for the optimal estimation of the functional Aξ of unknown values ξ(t), t ≥ 0, of the stochastic process ξ(t) with stationary nth increments. The increment ξ (n) (t, τ ) admits + , one-sided moving average representation. If νuv < νuv the density (53) which admits the canonical factorization (11) is the least favorable in the class Dvu . The function cφτ = cφτ (t) ∫ ∞is determined by equality (40), conditions (42) and −∞ |φ(λ)|2 dλ = 2πP0 . The minimaxrobust spectral characteristic is calculated by formulas (31), (32) substituting f (λ) by f 0 (λ). Consider the problem of the optimal estimation of the functional AT ξ. In this case the least favorable spectral density is determined by the relation { { 2 }} f 0 (λ) = max v(λ), min u(λ), s0τ,T (λ) . (54) ∫

T

s0τ,T (λ) = c

DτT (AT φτ )(t)eiλt dt 0

Theorem 10 If there exists a solution φ0 = φ0 (t), t ∈ [0; T ], of equation (51) or equation (52) which satT T+ isfies conditions ||φ0 ||2 = P0 and νuv P0 = νuv P0 = τ τ 0 2 ||DT AT W φ || , spectral density (43) is least favorable in the class Dvu for the optimal estimation of the functional AT ξ of unknown values ξ(t), t ∈ [0; T ], of the stochastic process ξ(t) with stationary nth increments. The increment ξ (n) (t, τ ) admits one-sided moving avT T+ erage representation. If νuv < νuv , the density (54), which admits the canonical factorization (11), is least favorable in the class Dvu . The function cφτ = cφτ (t), t ∈ [0; T ],∫ is determined by equation (40), conditions ∞ (44) and −∞ |φ(λ)|2 dλ = 2πP0 . The minimax-robust spectral characteristic is calculated by formulas (35), (36) substituting f (λ) by f 0 (λ).

7

Least favorable spectral densities in the class Dδ

Consider the problem of the optimal estimation of the functionals Aξ and AT ξ of unknown values ξ(t), t ≥ 0, of the stochastic process ξ(t) with stationary nth increments in the case where the spectral density is not known, but the following set of spectral densities is given { } ∫ ∞ 1 Dδ = f (λ)| |f (λ) − v(λ)|dλ ≤ δ , 2π −∞ where v(λ) is a bounded spectral density. It comes from the condition 0 ∈ ∂∆Dδ (f 0 ) that the least favorable spectral densities in the class Dδ for optimal linear extrapolation of the functional Aξ can be presented in the form { ∫ ∞ 2 } f 0 (λ) = max v(λ), c Dτ (Aφ0τ )(t)eiλt dt , 0

(55) where unknown function cφ0τ (t) is calculated using the factorization (11), relation (40), condition (42) and condition ∫ ∞ ∫ ∞ 1 1 |φ0 (λ)|2 dλ = δ + v(λ)dλ = P1 . 2π −∞ 2π −∞ Define by νδ P1 the maximum value of ||Dτ AWτ φ||2 on the set of those φ which belongs to the set of solutions of equation (47), satisfy equation ||φ||2 = P1 , inequality ∫ ∞ 2 v(λ) ≤ φ(t)e−iλt dt 0

T Denote by νuv P0 the maximum value of τ τ 2 b T Wτ φ||2 on the set of ||DT AT W φ|| = ||DτT A those solutions φ of equations (51) and (52), which satisfy condition ||φ||2 = P0 , inequalities ∫ 2 T −iλt v(λ) ≤ φ(t)e dt ≤ u(λ) 0

and determine the canonical factorization (11) of the spectral density f (λ). Let νδ+ P1 be the maximum value of ||Dτ AWτ φ||2 on the set of those φ, which satisfy condition ||φ||2 = P1 and determine the canonical factorization (11) of the spectral density f 0 (λ) defined by (55). The following statement holds true.

and define the canonical factorization (11) of the specT+ tral density f (λ). Let νuv P0 be the maximum value τ τ 2 of ||DT AT W φ|| on the set of those φ which satisfy condition ||φ||2 = P0 and define canonical factorization (11) of the spectral density f 0 (λ) determined by (54). The following statement holds true.

Theorem 11 If there exists a solution φ0 = φ0 (t) of equation (47) which satisfies conditions ||φ0 ||2 = P1 and νδ P0 = νδ+ P1 = ||Dτ AWτ φ0 ||2 , the spectral density (41) is the least favorable in the class Dδ for the optimal extrapolation of the functional Aξ of unknown values ξ(t), t ≥ 0, of the stochastic process ξ(t) with stationary

Mathematics and Statistics 2(2): 78-88, 2014

nth increments. The increment ξ (n) (t, τ ) admits onesided moving average representation. If νδ < νδ+ , the density (55), which admits the canonical factorization (11), is least favorable in the class Dδ . The function cφτ = cφτ (t) is determined by equality (40), condition (42) and condition ∫ ∞ ∫ ∞ 1 1 |φ(λ)|2 dλ = δ + v(λ)dλ = P1 . (56) 2π −∞ 2π −∞ The minimax-robust spectral characteristic is calculated by formulas (31), (32) substituting f (λ) by f 0 (λ). In the case of optimal estimation of the functional AT ξ the least favorable spectral density is determined by formula  2  ∫   T f 0 (λ) = max v(λ), c DτT (AT φ0τ )(t)eiλt dt .   0 (57) Let νδT P1 be the maximum value of ||DτT AT Wτ φ||2 = b T Wτ φ||2 on the set of those φ which belong to ||DτT A the set of solutions of equation (51) and (52), satisfy condition ||φ||2 = P1 , the inequality ∫ 2 T −iλt v(λ) ≤ φ(t)e dt 0 and determined the canonical factorization (11) of the spectral density f (λ). Let νδT + P1 be the maximum value of ||DτT AT Wτ φ||2 on the set of those φ which satisfy condition ||φ||2 = P1 and determined the canonical factorization (11) of the spectral density f 0 (λ) defined by (57). The following statement holds true. Theorem 12 If there exists a solution φ0 = φ0 (t), t ∈ [0; T ], of equation (51) or equation (52) which satisfies conditions ||φ0 ||2 = P1 and νδT P1 = νδT + P1 = ||DτT AT Wτ φ0 ||2 , the spectral density (43) is least favorable in the class Dδ for the optimal extrapolation of the functional AT ξ of unknown values ξ(t), t ∈ [0; T ], of the stochastic process ξ(t) with stationary nth increments. The increment ξ (n) (t, τ ) admits one-sided moving average representation. If νδT < νδT + , the density (57) which admits the canonical factorization (11) is the least favorable in the class Dδ . The function cφτ = cφτ (t), t ∈ [0; T ], is determined by equation (40), conditions (44) and (56). The minimax-robust spectral characteristic is calculated by formulas (35), (36) substituting f (λ) by f 0 (λ).

8

In this article methods of solution of the problem of optimal linear estimation of functionals ∫

∫

∞

a(t)ξ(t)dt, 0

process ξ(t) for t < 0. Formulas are derived for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimates of functionals in the case of spectral certainty where the spectral density of the process is known. In the case of spectral uncertainty where the spectral density is not known but, instead, a set of admissible spectral densities is specified, the minimax-robust method was applied. We propose a representation of the mean square error in the form of a linear functional in L1 with respect to spectral density, which allows us to solve the corresponding conditional extremum problem and describe the minimax (robust) estimates of the functionals. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of the functionals are derived in this case for some concrete classes of admissible spectral densities.

REFERENCES [1] A. N. Kolmogorov. Selected works of A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics. Ed. by A. N. Shiryayev, Mathematics and Its Applications. Soviet Series. 26. Dordrecht etc.: Kluwer Academic Publishers, 1992. [2] N. Wiener. Extrapolation, interpolation and smoothing of stationary time series. Whis wngineering applications, The M. I. T. Press, Massachusetts Institute of Technology, Cambridge, Mass, 1966. [3] A. M. Yaglom. Correlation theory of stationary and related stochastic functions. Vol. 1: Basic results, Springer Series in Statistics, Springer-Verlag, New York etc., 1987. [4] A. M. Yaglom. Correlation theory of stationary and related stochastic functions. Vol. 2: Suplementary notes and references, Springer Series in Statistics, Springer-Verlag, New York etc., 1987. [5] Yu. A. Rozanov. Stationary stochastic processes. 2nd rev. ed, “Nauka”. Moskva, 1990. [6] A. M. Yaglom. Correlation theory of stationary and related stochastic processes with stationary nth increments, Mat. Sbornik, Vol.37, No.1, 141-196, 1955. [7] A. M. Yaglom. Some clases of stochastic fields in ndimentional space related with stochastic stationary processes, Teor. Veroyatn. Primen, Vol.2, 292-338, 1957.

Conclusions

Aξ =

87

AT ξ =

T

[8] M. S. Pinsker. The theory of curves with nth stationary incremens in Hilber spaces, Izvestiya Akademii Nauk SSSR. Ser. Mat., Vol.19, No.5, 319-344, 1955.

a(t)ξ(t)dt 0

which depend on unknown values of a stochastic process ξ(t) with stationary nth increments are described. The received estimates are based on observations of the

[9] M. S. Pinsker, A. M. Yaglom. On linear extrapolaion of stochastic processes with nth stationary incremens, Doklady Akademii Nauk SSSR, Vol.94, 385388, 1954.

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Robust Extrapolation Problem for Stochastic Processes with Stationary Increments

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[12] U. Grenander. A prediction problem in game theory, Arkiv f¨ or Matematik No.3, 371-379, 1957

[13] J. Franke, H. V. Poor. Minimax-robust filtering and finite-length robust predictors, In: Robust and Nonlinear Time Series Analysis, Lecture Notes in Statistics, Springer-Verlag, No.26, 87-126, 1984.

[14] J. Franke. Minimax robust prediction of discrete time series, Z. Wahrscheinlichkeitstheor. Verw. Gebiete, No.68, 337-364, 1985

[15] O. M. Kurkin, Yu. V. Korobochkin, S. A. Shatalov. Minimax information processing, Energoatomizdat, Moskva, 1990.

[16] M. P. Moklyachuk. Robust estimations of functionals of stochastic processes, Vydavnycho-Poligrafichnyi Tsentr, Kyivskyi Universytet, Kyiv, 2008.

[17] M. P. Moklyachuk, O. Yu. Masyutka. Minimaxrobust estimation technique for stationary stochastic processes. LAP LAMBERT Academic Publishing, 2012.

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