minimax prediction of random processes with stationary ... · minimax prediction of random...

Luz & Moklyachuk, Cogent Mathematics (2016), 3: 1133219http://dx.doi.org/10.1080/23311835.2015.1133219

STATISTICS | RESEARCH ARTICLE

Minimax prediction of random processes with stationary increments from observations with stationary noiseMaksym Luz1 and Mikhail Moklyachuk1*

Abstract: We deal with the problem of mean square optimal estimation of linear functionals which depend on the unknown values of a random process with stationary increments based on observations of the process with noise, where the noise process is a stationary process. Formulas for calculating values of the mean square errors and the spectral characteristics of the optimal linear estimates of the functionals are derived under the condition of spectral certainty, where the spectral densities of the processes are exactly known. In the case of spectral uncertainty, where the spectral densities of the processes are not exactly known while a class of admissible spectral densities is given, relations that determine the least favorable spectral densities and the minimax robust spectral characteristics are proposed.

Subjects: Mathematics Statistics; Operations Research; Optimization; Probability; Science; Statistical Theory Methods; Statistics; Statistics Probability; Stochastic Models Processes

Keywords: random process with stationary increments; minimax robust estimate; mean square error; least favorable spectral density; minimax spectral characteristic

AMS Subject classifications: 60G10; 60G25; 60G35; Secondary: 62M20; 93E10; 93E11

*Corresponding author: Mikhail Moklyachuk, Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine Email: [email protected]

Reviewing editor: Zudi Liu, University of Southampton, UK

Additional information is available at the end of the article

Received: 22 October 2015Accepted: 14 December 2015First Published: 19 January 2016

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.

of 17

ABOUT THE AUTHORSMaksym Luz is a Ph D student, Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv. His research interests include estimation problems for random processes and sequences with stationary increments. Mikhail Moklyachuk is a Professor, Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv. He received Ph D degree in Physics and Mathematical Sciences from the Taras Shevchenko University of Kyiv in 1977. His research interests are statistical problems for stochastic processes and random fields. Member of editorial boards of several international journals

PUBLIC INTEREST STATEMENTThe crucial assumption of most of papers dedicated to the problem of estimating the unobserved values of random processes is that spectral densities of the involved processes are exactly known. However, the established results cannot be directly applied to practical estimation problems, because complete information of the spectral densities is impossible in most cases. This is a reason to derive the minimax estimates since they minimize the maximum of the mean-square errors for all spectral densities from a given set of admissible spectral densities simultaneously. In this article we deal with the problem of optimal estimation of functionals depending on unknown values of random processes from observations of the process with noise in the case of spectral uncertainty where spectral densities are not known exactly while a class of admissible spectral densities is given. Formulas that determine least favourable spectral densities and minimax (robust) spectra characteristics are derived.

http://crossmark.crossref.org/dialog/?doi=10.1080/23311835.2015.1133219&domain=pdf&date_stamp=2016-01-19

mailto:[email protected]

http://creativecommons.org/licenses/by/4.0/

of 17


1. IntroductionTraditional methods of finding solutions to problems of estimation of unobserved values of a random process based on a set of available observations of this process, or observations of the process with a noise process, are developed under the condition of spectral certainty, where the spectral densities of the processes are exactly known. Methods of solution of these problems, which are known as in-terpolation, extrapolation, and filtering of stochastic processes, were developed for stationary sto-chastic processes by A.N. Kolmogorov, N. Wiener, and A.M. Yaglom (see selected works by Kolmogorov (1992), books by Wiener (1966), Yaglom (1987a, 1987b), Rozanov (1967). Stationary stochastic pro-cesses and sequences admit some generalizations, which are properly described in books by Yaglom (1987a, 1987b). Random processes with stationary nth increments are among such generalizations. These processes were introduced in papers by Pinsker and Yaglom (1954), Yaglom (1955, 1957), and Pinsker (1955). In the indicated papers, the authors described the spectral representation of the stationary increment process and the canonical factorization of the spectral density, solved the ex-trapolation problem, and proposed some examples.

Traditional methods of finding solutions to extrapolation, interpolation, and filtering problems may be employed under the basic assumption that the spectral densities of the considered random processes are exactly known. In practice, however, the developed methods are not applicable since the complete information on the spectral structure of the processes is not available in most cases. To solve the problem, the parametric or nonparametric estimates of the unknown spectral densities are found or these densities are selected by other reasoning. Then, the classical estimation method is applied, provided that the estimated or selected densities are the true ones. However, as was shown by Vastola and Poor (1983) with the help of concrete examples, this method can result in significant increase of the value of the error of estimate. This is a reason to search estimates which are optimal for all densities from a certain class of the admissible spectral densities. The introduced 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. The paper by Grenander (1957) should be marked as the first one where the minimax approach to extrapolation problem for stationary processes was proposed. Franke and Poor (1984) and Franke (1985) investi-gated 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. A survey of results in minimax (robust) methods of data processing can be found in the paper by Kassam and Poor (1985). A wide range of results in minimax robust extrapolation, interpolation, and filtering of random pro-cesses and sequences belong to Moklyachuk (2000, 2001, 2008a) . Later, Moklyachuk and Masyutka (2011 – 2012) developed the minimax technique of estimation for vector-valued stationary pro-cesses and sequences. Dubovets’ka, Masyutka, and Moklyachuk (2012) investigated the problem of minimax robust interpolation for another generalization of stationary processes—periodically cor-related sequences. In the further papers, Dubovets’ka and Moklyachuk (2013a, 2013b, 2014a, 2014b) investigated the minimax robust extrapolation, interpolation, and filtering problems for periodically correlated processes and sequences.

The minimax robust extrapolation, interpolation, and filtering problems for stochastic sequences with nth stationary increments were investigated by Luz and Moklyachuk (2012,, 2013a, 2013b, 2014a, 2014b, 2015a, 2015b, 2015c); Moklyachuk and Luz (2013). In particular, the minimax robust extrapolation problem based on observations with and without noise for such sequences is investi-gated in papers by Luz and Moklyachuk (2015b), Moklyachuk and Luz (2013). Same estimation prob-lems for random processes with stationary increments with continuous time are investigated in articles by Luz and Moklyachuk (2014a, 2015a, 2015b).

In this article, we deal with the problem of the mean square optimal estimation of the linear func-tionals A� = ∫∞

0a(t)�(t)dt and AT� = ∫ T

0a(t)�(t)dt which depend on the unknown values of a ran-

dom process �(t) with stationary nth increments from observations of the process �(t) + �(t) at points t < 0, where �(t) is an uncorrelated with �(t) stationary process. The case of spectral certainty

of 17


as well as the case of spectral uncertainty are considered. Formulas for calculating values of the mean square errors and the spectral characteristics of the optimal linear estimates of the function-als are derived under the condition of spectral certainty, where the spectral densities of the pro-cesses are exactly known. In the case of spectral uncertainty, where the spectral densities of the processes are not exactly known while a class of admissible spectral densities is given, relations that determine the least favorable spectral densities and the minimax spectral characteristics are de-rived for some classes of spectral densities.

2. Stationary random increment process. Spectral representationIn this section, we present basic definitions and spectral properties of random processes with sta-tionary increment. For more details, see the book by Yaglom (1987a, 1987b).

Definition 2.1 For a given random process �(t), t ∈ ℝ, the process

where B� is a backward shift operator with a step � ∈ ℝ, such that B

��(t) = �(t − �) is called the ran-

dom nth increment with step � ∈ ℝ generated by the random process �(t).

Definition 2.2 The random nth increment process �(n)(t, �) generated by a random process �(t), t ∈ ℝ, is in wide sense stationary, if the mathematical expectations

exist for all t0, �, t, �

1, �2 and do not depend on t

0. The function c(n)(�) is called the mean value of the

nth increment and the function D(n)(t, �1, �2) is called the structural function of the stationary nth

increment (or the structural function of nth order of the random process �(t), t ∈ ℝ).

The random process �(t), t ∈ ℝ, which determines the stationary nth increment process �(n)(t, �) by formula (1) is called the process with stationary nth increments.

The following theorem describes representations of the mean value and the structural function of the random stationary nth increment process �(n)(t, �).

Theorem 2.1 The mean value c(n)(�) and the structural function D(n)(t, �1, �2) of the random station-

ary nth increment process �(n)(t, �) can be represented in the following forms:

where c is a constant and F(�) is a left-continuous nondecreasing bounded function, such that F(−∞) = 0. The constant c and the function F(�) are determined uniquely by the increment process �(n)(t, �).

The representation (3) of the structural function D(n)(t, �1, �2) and the Karhunen theorem (see

Karhunen, 1947) allow us to write the following spectral representation of the stationary nth incre-ment process �(n)(t, �):

where Z�(n)(�) is a random process with uncorrelated increments on ℝ connected with the spectral

function F(�) from representation (3) by the relation

(1)�(n)(t, �) = (1 − B�)n�(t) =

n∑l=0

(−1)l(n

l

)�(t − l�),

��(n)(t0, �) = c(n)(�),

��(n)(t0+ t, �

1)�(n)(t

0, �2) = D(n)(t, �

1, �2)

(2)c(n)(�) = c�n,

D(n)(t, �1, �2) = ∫

∞

−∞

ei�t(1 − e−i�1�)n(1 − ei�2�)n(1 + �2)n

�2ndF(�),

(4)�(n)(t, �) = ∫∞

−∞

eit�(1 − e−i��)n(1 + i�)n

(i�)ndZ

�(n)(�),

(3)

of 17


3. The Hilbert space projection method of extrapolationConsider a random process �(t), t ∈ ℝ, which generates a stationary random increment process �(n)(t, �) with the absolutely continuous spectral function F(�) and the spectral density function f (�). Let �(t), t ∈ ℝ, be another random process which is stationary and uncorrelated with �(t). Suppose that the process �(t) has absolutely continuous spectral function G(�) and the spectral density g(�). Without loss of generality, we can assume that the increment step 𝜏 > 0 and both processes �(n)(t, �) and �(t) have zero mean values: ��(n)(t, �) = 0, ��(t) = 0.

The main purpose of this paper is to find optimal, in the mean square sense, linear estimates of the functionals

which depend on the unknown values of the random process �(t) at time t ≥ 0 based on observa-tions of the process � (t) = �(t) + �(t) at time t < 0.

For further analysis, we need to make the following assumptions. Let the function a(t), t ≥ 0, which determines the functionals A�, AT�, and the linear transformation D�, being defined below, satisfy the conditions

and

Suppose also that the spectral densities f (�) and g(�) satisfy the minimality condition

for some function �(�) of the form = ∫∞0�(t)ei�tdt. Assumption (8) guarantees that the mean square

errors of estimates of the considered functionals are greater than 0.

Following the classical estimation theory developed for stationary processes, it is reasonable to apply the method proposed by Kolmogorov (see selected works by Kolmogorov (1992)), where the estimate is a projection of an element of the Hilbert space H = L

2(Ω,�, �) of the random variables �

with zero mean value, �� = 0, and finite variance, �|𝛾|2 < ∞ on a subspace of the space H = L

2(Ω,�, �). The inner product in the space H = L

2(Ω,�, �) is defined as (�

1; �

2) = ��

1�2. Since

we have no observations of the process �(t) to take as initial values, the issue is that both functionals A� and AT� have infinite variance. Thus, we need to derive other objects from the space H = L

2(Ω,�, �)

to proceed with the Hilbert space projection method.

Consider a representation of the functional A� in the form

where

(5)�|Z𝜉(n)(t2) − Z

𝜉(n)(t1)|2 = F(t

2) − F(t

1) < ∞ for all t

2> t

1, t

1∈ ℝ, t

2∈ ℝ.

A� = ∫∞

0

a(t)�(t)dt, AT� = ∫T

0

a(t)�(t)dt

(6)∫∞

0

|a(t)|dt < ∞, ∫∞

0

t|a(t)|2dt < ∞,

(7)∫∞

0

|D𝜏�(t)|dt < ∞, ∫

∞

0

t|D𝜏�(t)|2dt < ∞.

(8)∫∞

−∞

|𝛾(𝜆)|2𝜆2n|1 − ei𝜆𝜏 |2n(1 + 𝜆2)n((1 + 𝜆2)nf (𝜆) + 𝜆2ng(𝜆))

d𝜆 < ∞,

A� = A� − A�,

A� = ∫∞

0

a(t)� (t)dt, A� = ∫∞

0

a(t)�(t)dt.

of 17


Under the condition (6), the functional A� has finite variance and, hence, it belongs to the space H = L

2(Ω,�, �). A representation of the functional A� is described in the following lemma.

Lemma 3.1 The linear functional A� admits a representation

where

[x]� denotes the least integer number among numbers that are greater than or equal to x, coefficients {d(k):k ≥ 0} are determined from the relation

D� is a linear transformation of a function x(t), t ≥ 0, defined by the formula

Corollary 3.1 The linear functional AT� admits a representation

where

D�T is a linear transformation of an arbitrary function x(t), t ∈ [0; T], defined by the formula

Under the condition (7), the functional B� from Lemma 3.1 belongs to the space H = L2(Ω,�, �),

while the functional V� is observed and can be considered as an initial value. Thus, Lemma 3.1 im-plies the following representation of the functional A�:

A� = B� − V� ,

(9)

B� = �∞

0

b�(t)� (n)(t, �)dt, V� = �

0

−�n

v�(t)�(t)dt,

v�(t) =

n∑l=[−t

�

]�(−1)l

(n

l

)b�(t + l�), t ∈ [−�n; 0),

b�(t) =

∞∑k=0

a(t + �k)d(k) = D��(t), t ≥ 0,

∞∑k=0

d(k)xk =

(∞∑j=0

xj

)n

,

D��(t) =

∞∑k=0

x(t + �k)d(k).

AT� = BT� − VT� ,

(11)

BT� = ∫T

0

b�,T(t)�

(n)(t, �)dt, VT� = ∫0

−�n

v�,T(t)�(t)dt,

v�,T(t) =

min{[

T−t

�

],n}

∑l=[−t

�

]�(−1)l

(n

l

)b�,T(t + l�), t ∈ [−�n;0),

b�,T(t) =

[T−t

�

]∑k=0

a(t + �k)d(k) = D�T�(t), t ∈ [0;T], (12)

D�T�(t) =

[T−t

�

]∑k=0

x(t + �k)d(k).

A� = A� − A� = B� − A� − V� = H� − V� ,

(10)

of 17


where the functional H�: = B� − A� belongs to the space H = L2(Ω,�, �) and the Hilbert space

projection method can be applied. Since the functional V� depends on the observations � (t), −𝜏n ≤ t < 0, the following relations hold true for the estimates A�, H� and the mean square errors Δ(f ,g; A�), Δ(f ,g; H�):

Therefore, the problem is reduced to finding the optimal mean square estimate H� of the functional H�.

The next step is to describe the spectral structure of the functional H�. The stationary random process �(t) admits the spectral representation (see Gikhman & Skorokhod, 2004).

where Z�(�) is a random process with uncorrelated increments on ℝ which correspond to the spec-

tral function G(�). Taking into account (4), the spectral representation of the random process � (n)(t, �) can be described by the formulas

where

One can easily conclude that the spectral density p(�) of the random process � (t) is the following:

The functional H� admits the spectral representation

where

Denote by H0−(�(n)�

+ �(n)�) the closed linear subspace of the space H = L

2(Ω,�, �), which is gener-

ated by observations {𝜉(n)(t, 𝜏) + 𝜂(n)(t, 𝜏):t < 0}, 𝜏 > 0. Denote by L0−2(p) the closed linear subspace

of the Hilbert space L2(p) defined by the set of functions

It follows from the equality

that the operator which maps the vector

(13)A� = H� − V� ,

Δ(f ,g; A�): = �|A� − A�|2 = �|H� − V� − H� + V� |2 = �|H� − H�|2 = :Δ(f ,g;H�).

�(t) = ∫∞

−∞

ei�tdZ�(�),

� (n)(t, �) = ∫∞

−∞

ei�t(1 − e−i��)n(1 + i�)n

(i�)ndZ

�(n)(�) + ∫

∞

−∞

ei�t(1 − e−i��)n(1 + i�)n

(i�)ndZ

�(n)(�)

= ∫∞

−∞

ei�t(1 − e−i��)n(1 + i�)n

(i�)ndZ

�(n)(�) + ∫

∞

−∞

ei�t(1 − e−i��)ndZ�(�),

dZ�(n)(�) = (i�)n(1 + i�)−ndZ

�(�), � ∈ ℝ.

p(�) = f (�) +�2n

(1 + �2)ng(�).

H� = ∫∞

−∞

B�(�)(1 − e−i��)n

(1 + i�)n

(i�)ndZ

�(n)+�(n)(�) − ∫

∞

−∞

A(�)dZ�(�),

B�(�) = ∫

∞

0

b�(t)ei�tdt = ∫

∞

0

(D��)(t)ei�tdt, A(�) = ∫

∞

0

a(t)ei�tdt.

{ei𝜆t(1 − e−i𝜆𝜏)n(1 + i𝜆)n(i𝜆)−n:t < 0

}.

(14)�(n)(t, �) + �(n)(t, �) = ∫∞

−∞

ei�t(1 − e−i��)n(1 + i�)n

(i�)ndZ

�(n)+�(n)(�)

of 17


of the space L0−2(p) to the vector �(n)(t, �) + �(n)(t, �) of the space H0−(�(n)

�+ �(n)

�) may be extended

to a linear isometry between the above spaces. The following relation holds true:

Every linear estimate A� of the functional A� admits the representation

where h�(�) is the spectral characteristic of the estimate H�. We can find the estimate H� as a pro-

jection of the element H� of the space H on the subspace H0−(�(n)�

+ �(n)�). This projection is charac-

terized by two conditions:

(1) H� ∈ H0−(�(n)�

+ �(n)�);

(2) (H� − H�) ⟂ H0−(�(n)�

+ �(n)�). Condition (2) and property (15) imply the following relations

which hold true for every t < 0:

Let us define for � ∈ ℝ the function

and its Fourier transform

We have ��(t) = 0 for t < 0, hence

which allows us to construct the representation of the spectral characteristic

It follows from the condition 1) that the spectral characteristic h�(�) admits the representation

ei�t(1 − e−i��)n(1 + i�)n(i�)−n

(15)�� (n)(t1, �1)� (n)(t

2, �2) = ∫

∞

−∞

ei�(t1−t2)(1 − e−i��1 )n(1 − ei��2 )n(1 + �2)n

�2np(�)d�.

(16)A� = ∫∞

−∞

h�(�)dZ

�(n)+�(n)(�) − ∫

0

−�n

v�(t)(�(t) + �(t))dt,

(17)

�(H� − H�)(�(n)(t, �) + �(n)(t, �))

=1

2� ∫∞

−∞

[B�(�)(1 − e−i��)n − A(�) −

(i�)nh�(�)

(1 + i�)n

]e−i�t(1 − ei��)ng(�)d�

+1

2� ∫∞

−∞

[B�(�)(1 − e−i��)n −

(i�)nh�(�)

(1 + i�)n

]e−i�t(1 − ei��)n

(1 + �2)n

�2nf (�)d� = 0.

C�(�) =

[(B�(�)(1 − e−i��)n −

(i�)nh�(�)

(1 + i�)n

)(1 + �2)n

�2np(�) − A(ei�)g(�)

](1 − ei��)n

��(t) =

1

2� ∫∞

−∞

C�(�)e−i�td�, t ∈ ℝ.

C�(�) = ∫

∞

0

��(t)ei�tdt,

h�(�) = B

�(�)

(1 − e−i��)n(1 + i�)n

(i�)n− A(�)

(−i�)ng(�)

(1 − i�)np(�)−

(−i�)nC�(�)

(1 − ei��)n(1 − i�)np(�).

h�(�) = h(�)(1 − e−i��)n

(1 + i�)n

(i�)n,

h(�) = ∫0

−∞

s(t)ei�tdt, s(t) ∈ L−2,

of 17


which leads to the following relations holding true for every s ≥ 0:

Relation (17) can be represented in terms of linear operators in the space L2[0;∞). Let us define the

operators

where �(t), �(t), �(t) ∈ L2[0;∞). The introduced operators allow us to represent relations (17) in the

form

where

Then, under the condition that the linear operator �� is invertible, the function �

�(t), t ≥ 0, can be

found by the formula

Consequently, the spectral characteristic h�(�) of the optimal estimate H� of the functional H� is

calculated by the formula

where

The value of mean square error is calculated by the formula

(17)∫∞

−∞

[B�(�) −

A(�)(1 − e−i��)−n�2ng(�)

(1 + �2)np(�)−

|1 − ei�� |−n�2nC�(�)

(1 + �2)np(�)

]e−i�sd� = 0.

(��)(s) =

1

2� ∫∞

0

�(t) ∫∞

−∞

ei�(t−s)�2ng(�)

|1 − ei�� |2n(1 + �2)np(�)d�dt, s ∈ [0;∞),

(��)(s) =

1

2� ∫∞

0

�(t) ∫∞

−∞

ei�(t−s)�2n

|1 − ei�� |2n(1 + �2)np(�)d�dt, s ∈ [0;∞),

(��)(s) =1

2� ∫∞

0

�(t) ∫∞

−∞

ei�(t−s)f (�)g(�)

p(�)d�dt, s ∈ [0; ∞),

b�(s) − (�

��)(s) = (�

��)(s), s ≥ 0,

(18)��(t) =

min{n;[t

�

]}∑l=0

(−1)l(n

l

)a(t − �l), t ≥ 0.

��(t) = (�−1

�D�

� − �−1

��

��)(t), t ≥ 0.

(19)

h�(�) = B

�(�)

(1 − e−i��)n(1 + i�)n

(i�)n−A(�)(1 + i�)n(−i�)ng(�)

(1 + �2)nf (�) + �2ng(�)

−(1 + i�)n(−i�)nC

�(�)

(1 − ei��)n((1 + �2)nf (�) + �2ng(�)

) ,

C�(�) = ∫

∞

0

(�−1

�D�

� − �−1

��

��)(t)ei�tdt.

(20)

Δ(f , g; A�) = Δ(f , g; H�) = ��H� − H��2

=1

2� ∫∞

−∞

��A(�)(1 − ei��)n(1 + �2)nf (�) − �2nC

�(�)

��2

�1 − ei�� 2n(1 + �2)2n(f (�) + �2n

(1+�2)ng(�))2

g(�)d�

+1

2� ∫∞

−∞

��A(�)(1 − ei��)n(−i�)ng(�) + (−i�)nC

�(�)

��2

�1 − ei�� 2n(1 + �2)n(f (�) + �2n

(1+�2)ng(�))2

f (�)d�

= ⟨D�� − �

��,�−1

�D�

� − �−1

��

��⟩ + ⟨��, �⟩.

of 17


The obtained results can be summarized in the following theorem.

Theorem 3.1 Let �(t), t ∈ ℝ, be a random process with stationary nth increment process �(n)(t, �) and let �(t), t ∈ ℝ, be an uncorrelated with �(t) stationary random process. Suppose that the spectral den-sities f (�) and g(�) of the random processes �(t) and �(t) satisfy the minimality condition (8) and the function a(t), t ≥ 0, satisfies conditions (6) and (7). Suppose also that the linear operator �

� is invertible.

The optimal estimate A� of the functional A� based on observations �(t) + �(t) at time t < 0 is calcu-lated by formula (16). The spectral characteristic h

�(�) and the value of mean square error Δ(f , g;A�) of

the estimate A� can be calculated by formulas (19) and (20), respectively.

Remark 3.1 The spectral characteristic h�(�) determined by formula (19) can be presented in the

form h�(�) = h1

�(�) − h2

�(�), where

The functions h1�(�) and h2

�(�) are the spectral characteristics of the mean square optimal estimates

B� and A� of the functionals B� and A�, respectively, based on observations �(t) + �(t) at time t < 0.

In the case of observations without noise, we have the following corollary.

Corollary 3.2 Let �(t), t ∈ ℝ, be a random process with stationary nth increment process �(n)(t, �). Suppose that the spectral density f (�) of the random processes �(t) satisfies the minimality condition (8) with g(�) = 0 and the function a(t), t ≥ 0, satisfies conditions (6) and (7). Suppose also that the lin-ear operator �

� defined below is invertible. The optimal linear estimate A� of the functional A� which

depends on the unknown values �(t), t ≥ 0, of the random process �(t), based on observations of the process �(t), t < 0, is calculated by the formula

The spectral characteristic h��(�) and the mean square error Δ(f ;A�) of the optimal estimate A� of the

functional A� are calculated by the formulas

where �� is the linear operator in the space L

2[0;∞) determined by the formula

Remark 3.2 In Corollary 3.2, we provide formulas for calculating the optimal linear estimate A� of the functional A� and the value of the mean square error Δ(f ; A�) of the estimate A� based on observations

of the process �(t) at time t < 0 using the Fourier transform of the function �2n

|1 − ei�� |2n(1 + �2)nf (�). In

the article by Luz and Moklyachuk (2014a), the same problem is considered. However, a solution is

h1�(�) = B

�(�)

(1 − e−i�� )n(1 + i�)n

(i�)n−

(1 + i�)n(−i�)n ∫∞0(�−1

�D�

�)(t)ei�tdt

(1 − ei�� )n((1 + �2)nf (�) + �2ng(�)

) ,

h2�(�) = −

A(�)(1 + i�)n(−i�)ng(�)

(1 + �2)nf (�) + �2ng(�)−

(1 + i�)n(−i�)n ∫∞0(�−1

��

��)(t)ei�tdt

(1 − ei�� )n((1 + �2)nf (�) + �2ng(�)

) .

(21)A� = ∫∞

−∞

h��(�)dZ

�(n) (�) − ∫

0

−�n

v�(t)�(t)dt.

(22)h��(�) = B

�(�)

(1 − e−i��)n(1 + i�)n

(i�)n−

(−i�)n ∫∞0(�−1

�D�

�)(t)ei�tdt

(1 − ei��)n(1 − i�)nf (�),

Δ(f ; A�) =1

2� ∫∞

−∞

�2n��∫∞0 (�−1

�D�

�)(t)ei�tdt��2

�1 − ei�� 2n(1 + �2)nf (�)d� = ⟨�−1

�D�

�,D��⟩,

(��)(s) =

1

2� ∫∞

0

�(t) ∫∞

−∞

ei�(t−s)�2n

|1 − ei�� |2n(1 + �2)nf (�)d�dt, s ∈ [0; ∞).

(23)

of 17


derived in terms of the function ��(t), t ≥ 0, which is determined by the canonical factorization of the

function

Theorem (3.1) can be used to obtain the optimal estimate AT� of the functional AT� which depends on the unknown values �(t), 0 ≤ t ≤ T, of the random process �(t), based on observations of the process �(t) + �(t) at time t < 0. To derive the corresponding formulas, let us put �(t) = 0 if t > T. We get that the spectral characteristic h

�,T(�) of the optimal estimate

is calculated by the formula

where the linear operator ��T in .the space L

2[0;∞) is determined by the formula

the function ��,T(t), t ∈ [0; T + �n], is calculated by formula

The mean square error of the optimal estimate AT� is calculated by the formula

where the linear operator �T in the space L2[0;∞) is determined by the formula

|1 − e−i�� |2n(1 + �2)n

�2nf (�) =

||||∫∞

0

��(t)e−i�tdt

||||2

.

(24)AT� = ∫∞

−∞

h�,T(�)dZ�(n)+�(n) (�) − ∫

0

−�n

v�,T(t)(�(t) + �(t))dt,

(25)h�,T(�) = B

�T(�)

(1 − e−i��)n(1 + i�)n

(i�)n−AT(�)(1 + i�)

n(−i�)ng(�)

(1 + �2)nf (�) + �2ng(�)

−(1 + i�)n(−i�)nC�

T(�)

(1 − ei��)n((1 + �2)nf (�) + �2ng(�)

) ,

B�T(�) = ∫T

0

b�,T(t)e

i�tdt = ∫T

0

(D�T�T)(t)e

i�tdt,

AT(�) = ∫T

0

a(t)ei�tdt,

C�T(�) = ∫

∞

0

(�−1

�D�T�T − �

−1

��

�,T��,T)(t)ei�tdt,

(��T�)(s) =

1

2� ∫T+�n

0

�(t) ∫∞

−∞

e−i�(t+s)�2ng(�)

|1 − ei�� |2n(1 + �2)np(�)d�dt, s ∈ [0;∞),

��,T(t) =

min{[

t

�

],n}

∑

l=max

{[t−T

�

]�,0

}(−1)l

(n

l

)a(t − �l), 0 ≤ t ≤ T + �n.

(26)

Δ(f ,g; AT�) = Δ(f , g; HT�) = ��HT� − HT��2

=1

2� ∫∞

−∞

��AT(�)(1 − ei��)n(1 + �2)nf (�) − �2nC�

T(�)��2

�1 − ei�� 2n(1 + �2)2n(f (�) + �2n

(1+�2)ng(�))2

g(�)d�

+1

2� ∫∞

−∞

��AT(�)(1 − ei��)n(−i�)ng(�) + (−i�)nC�

T(�)��2

�1 − ei�� 2n(1 + �2)n(f (�) + �2n

(1+�2)ng(�))2

f (�)d�

= ⟨D�T�T − �

�,T��,T ,�−1

�D�T�T − �

−1

��

�,T��,T⟩ + ⟨�T�T , �T⟩,

of 17


and the function �T(t), t ∈ [0;T], is determined as �T(t) = a(t).

The described results can be summarized in the following theorem.

Theorem 3.2 Let �(t), t ∈ ℝ, be a random process with stationary nth increment process �(n)(t, �) and let �(t), t ∈ ℝ, be an uncorrelated with �(t) stationary random process. Suppose that the spectral den-sities f (�) and g(�) of the random processes �(t) and �(t) satisfy the minimality condition (8) and the function a(t), 0 ≤ t ≤ T, satisfies conditions (6) and (7). Suppose also that the linear operator �

� is

invertible. The optimal linear estimate AT� of the functional AT� based on observations of the process �(t) + �(t) at time t < 0 is calculated by formula (24). The spectral characteristic h

�,T(�) and the value of mean square error Δ(f , g; AT�) of the optimal estimate AT� are calculated by formulas (25) and (26), respectively.

4. Minimax robust method of extrapolationThe values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals A� and AT� based on observations of the process �(t) + �(t) or observations of the pro-cess �(t) without noise can be calculated by formulas (20), (23), (26) and (19), (22), and (25), respec-tively, in the case where the spectral densities f (�) and g(�) of the random processes �(t) and �(t) are exactly known. In the case where the spectral densities are not exactly known while sets = f ×g or = f of admissible spectral densities are given, the minimax robust method of estimation of the functionals which depend on the unknown values of the random process with stationary increments can be applied. The method consists in determining an estimate which mini-mizes the value of the mean square error for all spectral densities from the given class = f ×g or = f simultaneously. The following definitions formalize the proposed method.

Definition 4.1 For a given class of spectral densities, = f ×g spectral densities f0(�) ∈ f ,

g0(�) ∈ g are called the least favorable in the class for the optimal linear extrapolation of the

functional A� if the following relation holds true

Definition 4.2 For a given class of spectral densities = f ×g, the spectral characteristic h0(�) of the optimal linear estimate of the functional A� is called minimax robust if there are satisfied conditions:

Let us now formulate lemmas which follow from the introduced definitions and formulas (20) and (23) derived in the previous section.

Lemma 4.1 The spectral densities f 0(�) ∈ f and g0(�) ∈ g which satisfy the minimality condition (8) are the least favorable in the class for the optimal linear extrapolation of the functional A� based on observations of the random process �(t) + �(t) at time t < 0 if linear operators �0

�, �0

�, �0, determined by

the Fourier transform of the functions

determine a solution of the constrain optimization problem

(�T�)(s) =1

2� ∫T

0

�(t) ∫∞

−∞

ei�(t−s)(1 + �2)nf (�)g(�)

(1 + �2)nf (�) + �2ng(�)d�dt, s ∈ [0;∞),

Δ(f 0, g0) = Δ(h(f 0, g0); f 0, g0) = max(f ,g)∈f ×g

Δ(h(f , g); f , g).

h0(�) ∈ H =⋂

(f ,g)∈f ×g

L0−2(p(�)),

minh∈H

max(f ,g)∈f ×g

Δ(h; f , g) = max(f ,g)∈f ×g

Δ(h0; f , g).

�2n|1 − ei�� |−2n(1 + �2)nf 0(�) + �2ng0(�)

,�2n|1 − ei�� |−2ng0(�)

(1 + �2)nf 0(�) + �2ng0(�),

(1 + �2)nf 0(�)g0(�)

(1 + �2)nf 0(�) + �2ng0(�),

of 17


The minimax robust spectral characteristic h0 = h�(f 0, g0) can be found by formula (19) if

h�(f 0, g0) ∈ H.

The corresponding result holds true in the case where observations of the process �(t) at time t < 0 are available.

Lemma 4.2 The spectral density f 0 ∈ f satisfying the minimality condition

is the least favorable in the class f for the optimal linear extrapolation of the functional A� based on observations of the process �(t) at time t < 0 if the linear operator �0

� defined by the Fourier trans-

formation of the function

determines a solution to the constrain optimization problem

The minimax robust spectral characteristic h0 = h�(f 0) is calculated by formula (22) under the

condition h�(f 0) ∈ H.

The least favorable spectral densities can be found directly using the definition or applying the proposed lemmas. However, there is an approach which gives us a possibility to simplify the optimi-zation problem using the following property of the function Δ(h; f ,g). This function has a saddle point on the set H ×, which is formed by the minimax robust spectral characteristic h0 and a pair (f 0,g0) of the least favorable spectral densities. The saddle point inequalities

hold true if h0 = h�(f 0, g0), h

�(f 0,g0) ∈ H and the pair (f 0,g0) determines a solution of the

constrain optimization problem

where

In the case of estimating the functional A� based on the observations �(t), t < 0, we have the following constrain optimization problem

(27)

max(f ,g)∈f ×g

(⟨D�� − �

��,�−1

�D�

� − �−1

��

��⟩ + ⟨��, �⟩)

= ⟨D�� − �

0

��, (�0

�)−1D�

� − (�0�)−1�0

��⟩ + ⟨�0

�, �⟩.

(28)∫∞

−∞

|𝛾(𝜆)|2𝜆2n|1 − ei𝜆𝜏 |2n(1 + 𝜆2)nf (𝜆)

d𝜆 < ∞

�2n|1 − ei�� |−2n(1 + �2)−n(f 0(�))−1

(29)maxf∈f

⟨�−1

�D�

�,D��⟩ = ⟨(�0

�)−1D�

�,D��⟩.

Δ(h; f 0,g0) ≥ Δ(h0; f 0,g0) ≥ Δ(h0; f , g) ∀f ∈ f ,∀g ∈ g,∀h ∈ H

Δ(f ,g) = −Δ(h�(f 0,g0);f ,g) → inf, (f ,g) ∈ ,

Δ(h�(f 0, g0); f ,g) =

1

2� ∫∞

−∞

|||A(�)(1 − ei��)n(1 + �2)nf 0(�) − �2nC0

�(�)

|||2

|1 − ei�� |2n(1 + �2)2n(f 0(�) + �2n

(1+�2)ng0(�))2

g(�)d�

+1

2� ∫∞

−∞

|||A(�)(1 − ei��)n(−i�)ng0(�) + (−i�)nC0

�(�)

|||2

|1 − ei�� |2n(1 + �2)n(f 0(�) + �2n

(1+�2)ng0(�))2

f (�)d�,

C0�(ei�) = ∫

∞

0

((�0�)−1D�

� − (�0�)−1�0

��)(t)ei�tdt.

of 17


where

Using the indicator functions �(f ,g|f ×g) and �(f ,g|f ) of the sets f ×g and f , the indi-cated constrain optimization problems can be presented as unconditional optimization problems

respectively. In this case, solutions (f 0,g0) and f 0 are characterized by the conditions 0 ∈ �Δ(f0, g0)

and 0 ∈ �Δ(f0) which are necessary and sufficient conditions that the pair (f 0,g0) belongs to the

set of minimums of the convex functional Δ(f ,g) and the function f 0 belongs to the set of mini-mums of the convex functional Δ(f ). By �Δ(f

0, g0) and �Δ(f0), we denote subdifferentials of the

functionals Δ(f ,g) and Δ(f ) at point (f ,g) = (f 0,g0) and f 0, respectively (see books by Ioffe & Tihomirov, (1979), Moklyachuk, (2008a), Pshenichnyi, (1971), Rockafellar, (1997)).

5. Least favorable densities in the class 0

f ×0

g

In this section, we consider the problem of minimax robust extrapolation of the functional A� based on observations of the process �(t) + �(t) at time t < 0 on the set of admissible spectral densities = 0

f ×0

g, where

Let us suppose that the spectral densities f 0 ∈ 0

f , g0 ∈ 0

g and the functions

are bounded. These conditions ensure the functional Δ(h�(f 0,g0);f ,g) is continuous and bounded in

the space L1× L

1. Condition 0 ∈ �Δ(f

0, g0) implies the spectral densities f 0 ∈ 0

f , g0 ∈ 0

g satisfy the equalities

where the constants �1≥ 0, �

2≥ 0, and �

1≠ 0 if

�2≠ 0 if

We can summarize the obtained results in the following theorem.

Δ(f ) = −Δ(h�(f 0);f ) → inf, f ∈ f ,

Δ(h�(f 0);f ) =

1

2� ∫∞

−∞

�2n|||∫∞0 ((�0

�)−1D�

�)(t)ei�tdt|||2

|1 − ei�� |2n(1 + �2)n(f 0(�))2f (�)d�.

(30)

Δ(f ,g) = Δ(f ,g) + �(f , g|f ×g) → inf,

Δ(f ) = Δ(f ) + �(f |f ) → inf

0

f =

{f (�)| 1

2� �∞

−∞

f (�)d� ≤ P1

}, 0

g =

{g(�)| 1

2� �∞

−∞

g(�)d� ≤ P2

}.

(33)

h�,f (f

0, g0) =

|||A(�)(1 − ei��)n(−i�)ng0(�) + (−i�)nC0

�(�)

||||1 − ei�� |n(1 + �2)n∕2(f 0(�) + �2n

(1+�2)ng0(�))

,

h�,g(f

0, g0) =

|||A(�)(1 − ei��)n(1 + �2)nf 0(�) − �2nC0

�(�)

||||1 − ei�� |n(1 + �2)n(f 0(�) + �2n

(1+�2)ng0(�))

(32)|||A(�)(1 − e

i��)n(1 + �2)nf 0(�) − �2nC0�(�)

||| = �1|1 − ei�� |n

((1 + �2)nf 0(�) + �2ng0(�)

),

|||A(�)(1 − ei��)n(−i�)ng0(�) + (−i�)nC0

�(�)

||| = �2|1 − ei�� |n(1 + �2)−n∕2

((1 + �2)nf 0(�) + �2ng0(�)

),

1

2� ∫∞

−∞

f 0(�)d� = P1,

1

2� ∫∞

−∞

g0(�)d� = P2.

(31)

of 17


Theorem 5.1 Let the spectral densities f 0(�) ∈ 0

f and g0(�) ∈ 0

g satisfy condition (8) and let the functions h

�,f (f0, g0) and h

�,g(f0, g0) determined by formulas (30) and (31) be bounded. The spec-

tral densities f 0(�) and g0(�) are the least favorable in the class = 0

f ×0

g for the optimal linear extrapolations of the functional A� if they satisfy equations (32) and (33) and determine a solution of the optimization problem (27). The function h

�(f 0, g0) calculated by formula (19) is the minimax robust

spectral characteristic of the optimal estimate of the functional A�.

Theorem 5.2 Let the spectral density f (�) be known, let the spectral density g0(�) ∈ 0

g and let the spectral densities f (�), g0(�) satisfy the minimality condition (8). Suppose also that the function h�,g(f , g

0) determined by formula (31) is bounded. The spectral density

is the least favorable in the class 0

g for the optimal linear extrapolation of the functional A� if the func-tions f (�) + (1 + �2)−n�2ng0(�), g0(�) determine a solution of the optimization problem (27). The func-tion h

�(f , g0) calculated by formula (19) is the minimax robust spectral characteristic of the optimal

estimate of the functional A�.

Theorem 5.3 Let the spectral density g(�) be known, let the spectral density f 0(�) ∈ 0

f and let the spectral densities f 0(�), g(�) satisfy the minimality condition (8). Suppose also that the function h�,f (f

0, g) determined by formula (30) is bounded. The spectral density

is the least favorable in the class 0

f for the optimal linear extrapolation of the functional A� if the func-tion f 0(�) + (1 + �2)−n�2ng(�), determines a solution of the optimization problem (27). The function h�(f 0, g), calculated by formula (19), is the minimax robust spectral characteristic of the optimal esti-

mate of the functional A�.

In the case of estimating the functional A� based on the observations of the process �(t) at time t < 0 without noise, we can formulate the following theorem.

Theorem 5.4 Suppose that the spectral density f 0(�) ∈ 0

f satisfies condition (28). The spectral density

is the least favorable in the class = 0

f for the optimal linear extrapolations of the functional A� based on observations of the process �(t) at time t < 0 if it determines a solution of the optimization problem (29). The function h

�(f 0) calculated by formula (22) is the minimax robust spectral character-

istic of the optimal estimate of the functional A�.

(34)

g0(�) = max{0, f

1(�) − (1 + �2)n�−2nf (�)

},

f1(�) =

|||A(�)(1 − ei�� )n(1 + �2)nf (�) − �2nC0

�(�)

|||�1|1 − ei�� |n�2n ,

(35)

f 0(�) = max{0, g

2(�) − (1 + �2)−n�2ng(�)

},

g2(�) =

|||A(�)(1 − ei�� )n(−i�)ng(�) + (−i�)nC0

�(�)

|||�2|1 − ei�� |n(1 + �2)n∕2

,

f 0(�) =|�|n|||∫∞0 ((�0

�)−1D�

�)(t)ei�tdt|||

�1|1 − ei�� |n(1 + �2)n∕2

of 17


6. Least favorable densities in the class = u

v×

�

Let us consider the problem of minimax robust extrapolation of the functional A� based on observa-tions of the process �(t) + �(t) at time t < 0 on the set of admissible spectral densities = u

v ×�,

where

Here, the spectral densities u(�), v(�), and g1(�) are supposed to be known and the spectral densi-

ties u(�), v(�) are assumed to be bounded.

Using the condition 0 ∈ �Δ(f0, g0), we obtain the following equalities determining the spectral

densities: f 0 ∈ uv, g

0 ∈ �:

where the function �1(�) ≤ 0 and �

1(�) = 0 if f 0(�) ≥ v(�); the function �

2(�) ≥ 0 and �

2(�) = 0 if

f 0(�) ≤ u(�); and the function �(�) ≤ 0 and �(�) = 0 if g0(�) ≥ (1 − �)g1(�).

Theorem 6.1 Let the spectral densities f 0(�) ∈ uv and g0(�) ∈

� satisfy the minimality condition (8)

and let the functions h�,f (f

0, g0) and h�,g(f

0, g0) determined by formulas (30) and (31) be bounded. The spectral densities f 0(�) and g0(�) determined by equations (36) and (37) are the least favorable in the class = u

v ×� for the optimal linear extrapolations of the functional A� if they determine a solu-

tion of the optimization problem (27). The function h�(f 0, g0) calculated by formula (19) is the minimax

robust spectral characteristic of the optimal estimate of the functional A�.

Theorem 6.2 Let the spectral density f (�) be known, let the spectral density g0(�) ∈ � and let

the spectral densities f (�), g0(�) satisfy the minimality condition (8). Suppose also that the function h�,g(f , g

0) determined by formula (31) is bounded. The spectral density

where the function f1(�) is defined by formula (34), is the least favorable in the class

� for the optimal

linear extrapolation of the functional A� if the functions f (�) + (1 + �2)−n�2ng0(�), determine a solution of the optimization problem (27). The function h

�(f , g0) calculated by formula (19) is the minimax


Theorem 6.3 Let the spectral density g(�) be known, let the spectral density f 0(�) ∈ uv and let

the spectral densities f 0(�), g(�) satisfy the minimality condition (8). Suppose also that the function h�,f (f

0, g) determined by formula (30) is bounded. The spectral density

where the function g2(�) is defined by formula (35), is the least favorable in the class u

v for the optimal linear extrapolation of the functional A� if the function f 0(�) + (1 + �2)−n�2ng(�), determines a solution to optimization problem (27). The function h

�(f 0, g) calculated by formula (19) is the minimax


uv =

{f (�)|v(�) ≤ f (�) ≤ u(�), 1

2� ��

−�

f (�)d� = P1

},

�=

{g(�)|g(�) = (1 − �)g

1(�) + �w(�),

1

2� ��

−�

g(�)d� = P2

}.

(36)

|||A(�)(1 − ei��)n(1 + �2)nf 0(�) − �2nC0

�(�)

||| = |1 − ei�� |n((1 + �2)nf 0(�) + �2ng0(�)

)(�1(�) + �

2(�)

+ �1),|||A(�)(1 − e

i��)n(−i�)ng0(�) + (−i�)nC0�(�)

||| = |1 − ei�� |n(1 + �2)−n∕2((1 + �2)nf 0(�)

+�2ng0(�))(�(�) + �

2),

g0(�) = max{(1 − �)g

2(�), f

1(�) − (1 + �2)n�−2nf (�)

},

f 0(�) = min{u(�),max

{v(�),g

2(�) − (1 + �2)−n�2ng(�)

}},

(37)

of 17


In the case of estimating the functional A� based on the observations of the process �(t) at time t < 0 without noise, we can formulate the following theorem.

Theorem 6.4 Suppose that the spectral density f 0(�) ∈ uv satisfies condition (28). The spectral den-

sity

is the least favorable in the class = uv for the optimal linear extrapolations of the functional A�

based on observations of the process �(t) at time t < 0 if it determines a solution of the optimization problem (29). The function h

�(f 0) calculated by formula (22) is the minimax robust spectral charac-

teristic of the optimal estimate of the functional A�.

7. ConclusionsIn this paper, we present results of investigating of the problem of optimal linear estimation of the functionals A� = ∫∞

0a(t)�(t)dt and AT� = ∫ T

0a(t)�(t)dt which depend on the unknown values of a

random process �(t) with nth stationary increments based on observations of the process �(t) + �(t) at time t < 0. In the case where the spectral densities of the processes are known, we found formu-las for calculating the values of the mean square errors and the spectral characteristics of the esti-mates of the functionals A� and AT�. In the case where the spectral densities are not exactly known, but a set of admissible spectral densities was available, we applied the minimax robust method to derive relations which determine the least favorable spectral densities from the given set and the minimax robust spectral characteristics.

f 0(�) = min

⎧⎪⎨⎪⎩u(�),max

⎧⎪⎨⎪⎩v(�),

��n��∫∞0 ((�0�)−1D�

�)(t)ei�tdt��

�1�1 − ei�� n(1 + �2)n∕2

⎫⎪⎬⎪⎭

⎫⎪⎬⎪⎭

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

FundingThe author received no direct funding for this research.

Author detailsMaksym Luz1

Mikhail Moklyachuk1

E-mail: [email protected] ID: http://orcid.org/0000-0002-8260-15841 Department of Probability Theory, Statistics and Actuarial

Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, 01601, Ukraine.

Citation informationCite this article as: Minimax prediction of random processes with stationary increments from observations with stationary noise, Maksym Luz & Mikhail Moklyachuk, Cogent Mathematics (2016), 3: 1133219.

ReferencesDubovets’ka, I. I., Masyutka, O. Yu., & Moklyachuk, M. P.

(2012). Interpolation of periodically correlated stochastic sequences. Theory of Probability and Mathematical Statistics, 84, 43–56.

Dubovets’ka, I. I. & Moklyachuk, M. P. (2013a). Filtration of linear functionals of periodically correlated sequences. Theory of Probability and Mathematical Statistics, 86, 51–64.

Dubovets’ka, I. I. & Moklyachuk, M. P. (2013b). Minimax estimation problem for periodically correlated stochastic processes. Journal of Mathematics and System Science, 3(1), 26–30.

Dubovets’ka, I. I. & Moklyachuk, M. P. (2014a). Extrapolation of periodically correlated processes from observations

with noise. Theory of Probability and Mathematical Statistics, 88, 67–83.

Dubovets’ka, I. I., & Moklyachuk, M. P. (2014b). On minimax estimation problems for periodically correlated stochastic processes. Contemporary Mathematics and Statistics, 2, 123–150.

Franke, J. (1985). Minimax robust prediction of discrete time series. Z. Wahrsch. Verw. Gebiete, 68, 337–364.

Franke, J., & Poor, H. V. (1984). Minimax-robust filtering and finite-length robust predictors. Robust and nonlinear time series analysis, Lecture notes in statistics (Vol. 26, pp. 87–126). Heidelberg, Springer-Verlag.

Gikhman, I. I., & Skorokhod, A. V. (2004). The theory of stochastic processes. I. Berlin: Springer.

Golichenko, I. I., & Moklyachuk, M. P. (2014). Estimates of functionals of periodically correlated processes. Kyiv: NVP “Interservis”.

Grenander, U. (1957). A prediction problem in game theory. Arkiv för Matematik, 3, 371–379.

Ioffe, A. D., & Tihomirov, V. M. (1979). Theory of extremal problems (p. 460). Amsterdam, North-Holland Publishing Company.

Karhunen, K. (1947). Uber lineare Methoden in der Wahrscheinlichkeitsrechnung. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, 37, 3–79.

Kassam, S. A., & Poor, H. V. (1985). Robust techniques for signal processing: A survey. Proceedings of the IEEE, 73, 433–481.

Kolmogorov, A.N. (1992). Selected works of A. N. Kolmogorov. Volume II: Probability theory and mathematical statistics. Edited by A. N. Shiryayev. Dordrecht etc.: Kluwer Academic Publishers.

Luz, M. M., & Moklyachuk, M. P. (2012). Interpolation of functionals of stochastic sequences with stationary increments from observations with noise. Prykladna Statystyka. Aktuarna ta Finansova Matematyka, 2, 131–148.

mailto:[email protected]

http://orcid.org/0000-0001-6329-1413

of 17


© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.You are free to: Share — copy and redistribute the material in any medium or format Adapt — remix, transform, and build upon the material for any purpose, even commercially.The licensor cannot revoke these freedoms as long as you follow the license terms.

Under the following terms:Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.

Luz, M. M., & Moklyachuk, M. P. (2013a). Interpolation of functionals of stochastic sequences with stationary increments. Theory of Probability and Mathematical Statistics, 87, 117–133.

Luz, M. M., & Moklyachuk, M. P. (2013b). Minimax-robust filtering problem for stochastic sequence with stationary increments. Theory of Probability and Mathematical Statistics, 89, 117–131.

Luz, M., & Moklyachuk, M. (2014a). Robust extrapolation problem for stochastic processes with stationary increments. Mathematics and Statistics, 2, 78–88.

Luz, M., & Moklyachuk, M. (2014b). Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences. Statistics, Optimization & Information Computing, 2, 176–199.

Luz, M., & Moklyachuk, M. (2015a). Minimax interpolation problem for random processes with stationary increments. Statistics, Optimization & Information Computing, 3, 30–41.

Luz, M., & Moklyachuk, M. (2015b). Filtering problem for random processes with stationary increments. Contemporary Mathematics and Statistics, 3, 8–27.

Luz, M., & Moklyachuk, M. (2015c). Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences. Statistics, Optimization & Information Computing, 3, 160–188.

Moklyachuk, M., & Luz, M. (2013). Robust extrapolation problem for stochastic sequences with stationary increments. Contemporary Mathematics and Statistics, 1, 123–150.

Moklyachuk, M. P. (2000). Robust procedures in time series analysis. Theory Stoch. Process., 6(3–4), 127–147.

Moklyachuk, M. P. (2001). Game theory and convex optimization methods in robust estimation problems. Theory of Stochastic Processes, 7(1–2), 253–264.

Moklyachuk, M. P. (2008a). Robust estimates for functionals of stochastic processes. Kyiv: Kyiv University Publishing.

Moklyachuk, M. P. (2008b). Nonsmooth analysis and optimization (pp. 400), Kyiv: Kyivskyi Universitet.

Moklyachuk, M. (2015). Minimax-robust estimation problems for stationary stochastic sequences. Statistics, Optimization & Information Computing, 3, 348–419.

Moklyachuk, M. P., & Masyutka, O. Yu (2011). Minimax prediction problem for multidimensional stationary stochastic processes. Communications in Statistics – Theory and Methods, 40, 3700–3710.

Moklyachuk, M. P., & Masyutka, O. Yu. (2012). Minimax-robust estimation technique for stationary stochastic processes (pp. 296), SaarbrÜcken, LAP LAMBERT Academic Publishing.

Pinsker, M. S., & Yaglom, A. M. (1954). On linear extrapolation of random processes with nth stationary increments. Doklady Akademii Nauk SSSR, 94, 385–388.

Pinsker, M. S. (1955). The theory of curves with nth stationary increments in Hilbert spaces. Izvestiya Akademii Nauk SSSR. Ser. Mat., 19, 319–344.

Pshenichnyi, B. N. (1971). Necessary conditions for an extremum. Pure and Applied mathematics 4 (Vol. XVIII, p. 230), New York: Marcel Dekker, Inc.

Rockafellar, R. T. (1997). Convex Analysis (p. 451). Princeton, NJ: Princeton University Press.

Rozanov, Y. A. (1967). Stationary stochastic processes. San Francisco, CA: Holden-Day.

Vastola, K. S., & Poor, H. V. (1983). An analysis of the effects of spectral uncertainty on Wiener filtering. Automatica, 28, 289–293.

Wiener, N. (1966). Extrapolation, interpolation, and smoothing of stationary time series. With engineering applications. Massachusetts: The M. I. T. Press, Massachusetts Institute of Technology.

Yaglom, A. M. (1955). Correlation theory of stationary and related random processes with stationary nth increments. Mat. Sbornik, 37, 141–196.

Yaglom, A. M. (1957). Some classes of random fields in n-dimensional space related with random stationary processes. Teor. Veroyatn. Primen., 2, 292–338.

Yaglom, A. M. (1987a). Correlation theory of stationary and related random functions. Basic results (Vol. 1, p. 526). Springer series in statistics, New York (NY): Springer-Verlag.

Yaglom, A. M. (1987b). Correlation theory of stationary and related random functions. Supplementary notes and references (Vol. 2, p. 258). Springer Series in Statistics, New York (NY): Springer-Verlag.

minimax prediction of random processes with stationary ... · minimax prediction of random...

Documents