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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.
1. Introduction
Traditional 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 interpolation, extrapolation, and filtering of stochastic processes, were developed for stationary stochastic processes by A.N. Kolmogorov, N. Wiener, and A.M. Yaglom (see selected works by Kolmogorov (Kolmogorov (Citation1992)), books by Wiener (Citation1966), Yaglom (Citation1987a, Citationb), Rozanov (Citation1967)). Stationary stochastic processes and sequences admit some generalizations, which are properly described in books by Yaglom (Citation1987a, Citationb). Random processes with stationary nth increments are among such generalizations. These processes were introduced in papers by Pinsker and Yaglom (Citation1954), Yaglom (Citation1955, Citation1957), and Pinsker (Citation1955). 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 extrapolation 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 (Citation1983) 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 (Grenander (Citation1957)) should be marked as the first one where the minimax approach to extrapolation problem for stationary processes was proposed. Franke and Poor (Franke & Poor (Citation1984)) and Franke (Franke (Citation1985)) 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. A survey of results in minimax (robust) methods of data processing can be found in the paper by Kassam and Poor (Kassam & Poor (Citation1985)). A wide range of results in minimax robust extrapolation, interpolation, and filtering of random processes and sequences belong to Moklyachuk (Moklyachuk (Citation2000), Moklyachuk (Citation2001), Moklyachuk (Citation2008a), Citation2015). Later, Moklyachuk and Masyutka (2011–2012) developed the minimax technique of estimation for vector-valued stationary processes and sequences. Dubovets’ka, Masyutka, and Moklyachuk (Dubovets’ka et al. (Citation2012)) investigated the problem of minimax robust interpolation for another generalization of stationary processes—periodically correlated sequences. In the further papers, Dubovets’ka and Moklyachuk (Dubovets’ka & Moklyachuk (Citation2013a), Dubovets’ka & Moklyachuk (Citation2013b), Dubovets’ka & Moklyachuk (Citation2014a), Dubovets’ka & Moklyachuk (Citation2014b)) 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 (Luz (Citation2012), Luz & Moklyachuk (Citation2013a), Luz & Moklyachuk (Citation2013b), Luz & Moklyachuk (Citation2014a), Luz & Moklyachuk (Citation2014b), Luz & Moklyachuk (Citation2015a), Luz & Moklyachuk (Citation2015b), Luz & Moklyachuk (Citation2015c); Moklyachuk & Luz, Moklyachuk & Luz (Citation2013)). In particular, the minimax robust extrapolation problem based on observations with and without noise for such sequences is investigated in papers by Luz and Moklyachuk (Luz & Moklyachuk (Citation2015b)), Moklyachuk and Luz (Moklyachuk & Luz (Citation2013)). Same estimation problems for random processes with stationary increments with continuous time are investigated in articles by Luz and Moklyachuk (Luz & Moklyachuk (Citation2014a), Luz & Moklyachuk (Citation2015a), Luz & Moklyachuk (Citation2015b)).
In this article, we deal with the problem of the mean square optimal estimation of the linear functionals and
which depend on the unknown values of a random process
with stationary nth increments from observations of the process
at points
, where
is an uncorrelated with
stationary process. The case of spectral certainty 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 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 spectral characteristics are derived for some classes of spectral densities.
2. Stationary random increment process. Spectral representation
In this section, we present basic definitions and spectral properties of random processes with stationary increment. For more details, see the book by Yaglom (Citation1987a, Citationb).
Definition 2.1
For a given random process ,
, the process
(1)
(1)
where is a backward shift operator with a step
, such that
is called the random nth increment with step
generated by the random process
.
Definition 2.2
The random nth increment process generated by a random process
,
, is in wide sense stationary, if the mathematical expectations
exist for all and do not depend on
. The function
is called the mean value of the nth increment and the function
is called the structural function of the stationary nth increment (or the structural function of nth order of the random process
,
).
The random process ,
, which determines the stationary nth increment process
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 .
Theorem 2.1
The mean value and the structural function
of the random stationary nth increment process
can be represented in the following forms:
(2)
(2)
where c is a constant and is a left-continuous nondecreasing bounded function, such that
. The constant c and the function
are determined uniquely by the increment process
.
The representation (3) of the structural function and the Karhunen theorem (see Karhunen, Karhunen (Citation1947)) allow us to write the following spectral representation of the stationary nth increment process
:
(3)
(3)
where is a random process with uncorrelated increments on
connected with the spectral function
from representation (3) by the relation
(4)
(4)
3. The Hilbert space projection method of extrapolation
Consider a random process ,
, which generates a stationary random increment process
with the absolutely continuous spectral function
and the spectral density function
. Let
,
, be another random process which is stationary and uncorrelated with
. Suppose that the process
has absolutely continuous spectral function
and the spectral density
. Without loss of generality, we can assume that the increment step
and both processes
and
have zero mean values:
,
.
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 at time
based on observations of the process
at time
.
For further analysis, we need to make the following assumptions. Let the function a(t), , which determines the functionals
,
, and the linear transformation
, being defined below, satisfy the conditions
(5)
(5)
and(6)
(6)
Suppose also that the spectral densities and
satisfy the minimality condition
(7)
(7)
for some function of the form
. 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 (Kolmogorov (Citation1992))), where the estimate is a projection of an element of the Hilbert space of the random variables
with zero mean value,
, and finite variance,
on a subspace of the space
. The inner product in the space
is defined as
. Since we have no observations of the process
to take as initial values, the issue is that both functionals
and
have infinite variance. Thus, we need to derive other objects from the space
to proceed with the Hilbert space projection method.
Consider a representation of the functional in the form
where
Under the condition (6), the functional has finite variance and, hence, it belongs to the space
. A representation of the functional
is described in the following lemma.
Lemma 3.1
The linear functional admits a representation
where(8)
(8)
denotes the least integer number among numbers that are greater than or equal to x, coefficients
are determined from the relation
is a linear transformation of a function x(t),
, defined by the formula
Corollary 3.1
The linear functional admits a representation
where(9)
(9)
is a linear transformation of an arbitrary function x(t),
, defined by the formula
Under the condition (7), the functional from Lemma 3.1 belongs to the space
, while the functional
is observed and can be considered as an initial value. Thus, Lemma 3.1 implies the following representation of the functional
:
where the functional belongs to the space
and the Hilbert space projection method can be applied. Since the functional
depends on the observations
,
, the following relations hold true for the estimates
,
and the mean square errors
,
:
(10)
(10)
Therefore, the problem is reduced to finding the optimal mean square estimate of the functional
.
The next step is to describe the spectral structure of the functional . The stationary random process
admits the spectral representation (see Gikhman & Skorokhod, Gikhman & Skorokhod (Citation2004)).
where is a random process with uncorrelated increments on
which correspond to the spectral function
. Taking into account (4), the spectral representation of the random process
can be described by the formulas
where
One can easily conclude that the spectral density of the random process
is the following:
The functional admits the spectral representation
where
Denote by the closed linear subspace of the space
, which is generated by observations
,
. Denote by
the closed linear subspace of the Hilbert space
defined by the set of functions
It follows from the equality(11)
(11)
that the operator which maps the vector
of the space to the vector
of the space
may be extended to a linear isometry between the above spaces. The following relation holds true:
(12)
(12)
Every linear estimate of the functional
admits the representation
(13)
(13)
where is the spectral characteristic of the estimate
. We can find the estimate
as a projection of the element
of the space H on the subspace
. This projection is characterized by two conditions:
(1) |
| ||||
(2) |
|
Let us define for the function
and its Fourier transform
We have for
, hence
which allows us to construct the representation of the spectral characteristic
It follows from the condition 1) that the spectral characteristic admits the representation
which leads to the following relations holding true for every :
(15)
(15)
Relation (17) can be represented in terms of linear operators in the space . Let us define the operators
where . The introduced operators allow us to represent relations (17) in the form
where(16)
(16)
Then, under the condition that the linear operator is invertible, the function
,
, can be found by the formula
Consequently, the spectral characteristic of the optimal estimate
of the functional
is calculated by the formula
(17)
(17)
where
The value of mean square error is calculated by the formula(18)
(18)
The obtained results can be summarized in the following theorem.
Theorem 3.1
Let ,
, be a random process with stationary nth increment process
and let
,
, be an uncorrelated with
stationary random process. Suppose that the spectral densities
and
of the random processes
and
satisfy the minimality condition (8) and the function a(t),
, satisfies conditions (6) and (7). Suppose also that the linear operator
is invertible. The optimal estimate
of the functional
based on observations
at time
is calculated by formula (16). The spectral characteristic
and the value of mean square error
of the estimate
can be calculated by formulas (19) and (20), respectively.
Remark 3.1
The spectral characteristic determined by formula (19) can be presented in the form
, where
The functions and
are the spectral characteristics of the mean square optimal estimates
and
of the functionals
and
, respectively, based on observations
at time
.
In the case of observations without noise, we have the following corollary.
Corollary 3.2
Let ,
, be a random process with stationary nth increment process
. Suppose that the spectral density
of the random processes
satisfies the minimality condition (8) with
and the function a(t),
, satisfies conditions (6) and (7). Suppose also that the linear operator
defined below is invertible. The optimal linear estimate
of the functional
which depends on the unknown values
,
, of the random process
, based on observations of the process
,
, is calculated by the formula
(19)
(19)
The spectral characteristic and the mean square error
of the optimal estimate
of the functional
are calculated by the formulas
(20)
(20)
where is the linear operator in the space
determined by the formula
Remark 3.2
In Corollary 3.2, we provide formulas for calculating the optimal linear estimate of the functional
and the value of the mean square error
of the estimate
based on observations of the process
at time
using the Fourier transform of the function
. In the article by Luz and Moklyachuk (Luz & Moklyachuk (Citation2014a)), the same problem is considered. However, a solution is derived in terms of the function
,
, which is determined by the canonical factorization of the function
Theorem (3.1) can be used to obtain the optimal estimate of the functional
which depends on the unknown values
,
, of the random process
, based on observations of the process
at time
. To derive the corresponding formulas, let us put
if
. We get that the spectral characteristic
of the optimal estimate
(21)
(21)
is calculated by the formula(22)
(22)
where the linear operator in the space
is determined by the formula
the function ,
, is calculated by formula
The mean square error of the optimal estimate is calculated by the formula
(23)
(23)
where the linear operator in the space
is determined by the formula
and the function ,
, is determined as
.
The described results can be summarized in the following theorem.
Theorem 3.2
Let ,
, be a random process with stationary nth increment process
and let
,
, be an uncorrelated with
stationary random process. Suppose that the spectral densities
and
of the random processes
and
satisfy the minimality condition (8) and the function a(t),
, satisfies conditions (6) and (7). Suppose also that the linear operator
is invertible. The optimal linear estimate
of the functional
based on observations of the process
at time
is calculated by formula (24). The spectral characteristic
and the value of mean square error
of the optimal estimate
are calculated by formulas (25) and (26), respectively.
4. Minimax robust method of extrapolation
The values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals and
based on observations of the process
or observations of the process
without noise can be calculated by formulas (20), (23), (26) and (19), (22), and (25), respectively, in the case where the spectral densities
and
of the random processes
and
are exactly known. In the case where the spectral densities are not exactly known while sets
or
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 minimizes the value of the mean square error for all spectral densities from the given class
or
simultaneously. The following definitions formalize the proposed method.
Definition 4.1
For a given class of spectral densities, spectral densities
,
are called the least favorable in the class
for the optimal linear extrapolation of the functional
if the following relation holds true
Definition 4.2
For a given class of spectral densities , the spectral characteristic
of the optimal linear estimate of the functional
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 and
which satisfy the minimality condition (8) are the least favorable in the class
for the optimal linear extrapolation of the functional
based on observations of the random process
at time
if linear operators
,
,
, determined by the Fourier transform of the functions
determines a solution of the constrain optimization problem(24)
(24)
The minimax robust spectral characteristic can be found by formula (19) if
.
The corresponding result holds true in the case where observations of the process at time
are available.
Lemma 4.2
The spectral density satisfying the minimality condition
(25)
(25)
is the least favorable in the class for the optimal linear extrapolation of the functional
based on observations of the process
at time
if the linear operator
defined by the Fourier transformation of the function
determines a solution to the constrain optimization problem(26)
(26)
The minimax robust spectral characteristic is calculated by formula (22) under the condition
.
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 optimization problem using the following property of the function . This function has a saddle point on the set
, which is formed by the minimax robust spectral characteristic
and a pair
of the least favorable spectral densities. The saddle point inequalities
hold true if ,
and the pair
determine a solution of the constrain optimization problem
where
In the case of estimating the functional based on the observations
,
, we have the following constrain optimization problem
where
Using the indicator functions and
of the sets
and
, the indicated constrain optimization problems can be presented as unconditional optimization problems
(27)
(27) , respectively. In this case, solutions
and
are characterized by the conditions
and
which are necessary and sufficient conditions that the pair
belongs to the set of minimums of the convex functional
and the function
belongs to the set of minimums of the convex functional
. By
and
, we denote subdifferentials of the functionals
and
at point
and
, respectively (see books by Ioffe & Tihomirov, Ioffe & Tihomirov (Citation1979), Moklyachuk, 2008, Pshenichnyi, 1971, Rockafellar, 1997).
5. Least favorable densities in the class ![](//:0)
![](//:0)
In this section, we consider the problem of minimax robust extrapolation of the functional based on observations of the process
at time
on the set of admissible spectral densities
, where
Let us suppose that the spectral densities ,
and the functions
(28)
(28)
are bounded. These conditions ensure the functional is continuous and bounded in the space
. Condition
implies the spectral densities
,
satisfy the equalities
(29)
(29)
where the constants ,
, and
if
if
We can summarize the obtained results in the following theorem.
Theorem 5.1
Let the spectral densities and
satisfy condition (8) and let the functions
and
determined by formulas (30) and (31) be bounded. The spectral densities
and
are the least favorable in the class
for the optimal linear extrapolations of the functional
if they satisfy equations (32) and (33) and determine a solution of the optimization problem (27). The function
calculated by formula (19) is the minimax robust spectral characteristic of the optimal estimate of the functional
.
Theorem 5.2
Let the spectral density be known, let the spectral density
and let the spectral densities
,
satisfy the minimality condition (8). Suppose also that the function
determined by formula (31) is bounded. The spectral density
(30)
(30)
is the least favorable in the class for the optimal linear extrapolation of the functional
if the functions
,
determine a solution of the optimization problem (27). The function
calculated by formula (19) is the minimax robust spectral characteristic of the optimal estimate of the functional
.
Theorem 5.3
Let the spectral density be known, let the spectral density
and let the spectral densities
,
satisfy the minimality condition (8). Suppose also that the function
determined by formula (30) is bounded. The spectral density
(31)
(31)
is the least favorable in the class for the optimal linear extrapolation of the functional
if the function
determines a solution of the optimization problem (27). The function
, calculated by formula (19), is the minimax robust spectral characteristic of the optimal estimate of the functional
.
In the case of estimating the functional based on the observations of the process
at time
without noise, we can formulate the following theorem.
Theorem 5.4
Suppose that the spectral density satisfies condition (28). The spectral density
is the least favorable in the class for the optimal linear extrapolations of the functional
based on observations of the process
at time
if it determines a solution of the optimization problem (29). The function
calculated by formula (22) is the minimax robust spectral characteristic of the optimal estimate of the functional
.
6. Least favorable densities in the class ![](//:0)
![](//:0)
Let us consider the problem of minimax robust extrapolation of the functional based on observations of the process
at time
on the set of admissible spectral densities
, where
Here, the spectral densities ,
, and
are supposed to be known and the spectral densities
,
are assumed to be bounded.
Using the condition , we obtain the following equalities determining the spectral densities:
,
:
(32)
(32)
where the function and
if
; the function
and
if
; and the function
and
if
.
Theorem 6.1
Let the spectral densities and
satisfy the minimality condition (8) and let the functions
and
determined by formulas (30) and (31) be bounded. The spectral densities
and
determined by equations (36) and (37) are the least favorable in the class
for the optimal linear extrapolations of the functional
if they determine a solution of the optimization problem (27). The function
calculated by formula (19) is the minimax robust spectral characteristic of the optimal estimate of the functional
.
Theorem 6.2
Let the spectral density be known, let the spectral density
and let the spectral densities
,
satisfy the minimality condition (8). Suppose also that the function
determined by formula (31) is bounded. The spectral density
where the function is defined by formula (34), is the least favorable in the class
for the optimal linear extrapolation of the functional
if the functions
,
determine a solution of the optimization problem (27). The function
calculated by formula (19) is the minimax robust spectral characteristic of the optimal estimate of the functional
.
Theorem 6.3
Let the spectral density be known, let the spectral density
and let the spectral densities
,
satisfy the minimality condition (8). Suppose also that the function
determined by formula (30) is bounded. The spectral density
where the function is defined by formula (35), is the least favorable in the class
for the optimal linear extrapolation of the functional
if the function
determines a solution to optimization problem (27). The function
calculated by formula (19) is the minimax robust spectral characteristic of the optimal estimate of the functional
.
In the case of estimating the functional based on the observations of the process
at time
without noise, we can formulate the following theorem.
Theorem 6.4
Suppose that the spectral density satisfies condition (28). The spectral density
is the least favorable in the class for the optimal linear extrapolations of the functional
based on observations of the process
at time
if it determines a solution of the optimization problem (29). The function
calculated by formula (22) is the minimax robust spectral characteristic of the optimal estimate of the functional
.
7. Conclusions
In this paper, we present results of investigating of the problem of optimal linear estimation of the functionals and
which depend on the unknown values of a random process
with nth stationary increments based on observations of the process
at time
. In the case where the spectral densities of the processes are known, we found formulas for calculating the values of the mean square errors and the spectral characteristics of the estimates of the functionals
and
. 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.
Acknowledgements
The authors would like to thank the referees for careful reading of the article and giving constructive suggestions.
References
- Dubovets’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, Vol. 26, Lecture notes in statistics (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, New York, Oxford: 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. In A. N. Shiryayev (Ed.). Dordrecht: Kluwer Academic.
- Luz, M.M. & Moklyachuk, M.P. (2012). Interpolation of functionals of stochastic sequences with stationary. Prykl. Stat., Aktuarna Finans. Mat., (2), 131–148.
- 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(4), 348–419.
- Moklyachuk, M. P. & Masyutka, O Yu (2011). Minimax prediction problem for multidimensional stationary stochastic processes. Communications in Statistics -- Theory and Methods, 40(19–20), 3700–3710.
- Moklyachuk, M. P., & Masyutka, O. Y. (2012). Minimax-robust estimation technique for stationary stochastic processes (p. 296), Saarbr\"ucken: LAP LAMBERT Academic.
- 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(5), 319–344.
- Pshenichnyi, B.N. (1971). Necessary conditions for an extremum. Pure and Applied mathematics 4 (Vol. XVIII, p. 230), New York (NY): Marcel Dekker, Inc.
- Rockafellar, R. T. (1997). Convex Analysis (p. 451). Princeton, NJ: Princeton University Press.
- Rozanov, Yu A (1967). Stationary stochastic processes. San Francisco: 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(1), 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.