On Minimax Robust Estimation Problem for Stochastic Sequences with Harmonizable Stable Increments

1. Preliminaries and Problem Statement

1.1. Symmetric $\alpha$-Stable Random Sequence

In this subsection, we give a brief overview of properties of the symmetric $\alpha$-stable random sequences [3,4,16,17].

Definition 1  

(symmetric $\alpha$-stable random variable). A real random variable ξ is said to be symmetric α-stable, $S\alpha S$, if its characteristic function has the form $Eexp\left(it\xi \right)=exp(-c|t{|}^{\alpha })$ for some $c\ge 0$ and $0<\alpha \le 2.$ The real random variables ${\xi }_1,{\xi }_2,\dots ,{\xi }_n$ are jointly $S\alpha S$ if all linear combinations ${\sum }_{k=1}^n{a}_k{\xi }_k$ are $S\alpha S$, or, equivalently, if the characteristic function of the random vector $\overrightarrow{\xi }=({\xi }_1,\dots ,{\xi }_n)$ is of the form

$${\varphi }_{\overrightarrow{\xi }}\left(\overrightarrow{t}\right)=Eexp\left(i{\displaystyle \sum _{k=1}^n}{t}_k{\xi }_k\right)=exp\left\{-{\int }_{S_n}{\left|{\displaystyle \sum _{k=1}^n}{t}_k{x}_k\right|}^{\alpha }d{\Gamma }_{\overrightarrow{\xi }}\left(\overrightarrow{x}\right)\right\},$$

where $\overrightarrow{t}=(t_1,\dots ,t_n),t_1,\dots ,t_n$ are real numbers, and ${\Gamma }_{\overrightarrow{\xi }}\left(\overrightarrow{x}\right)$ is a symmetric measure defined on the unit sphere $S_n\in R^n$, called the spectral measure of the random vector $\overrightarrow{\xi }=({\xi }_1,\dots ,{\xi }_n)$. There is a one-to-one correspondence between the distribution of $\overrightarrow{\xi }$ and its spectral measure ${\Gamma }_{\overrightarrow{\xi }}\left(\overrightarrow{x}\right)$. (Cambanis [3]).

For real jointly $S\alpha S$ random variables $\xi ,\eta$ with $1<\alpha \le 2$ the covariation of $\xi ,\eta$ is defined as

$${[\xi ,\eta ]}_{\alpha }={\int }_{S_2}\left(x\right){\left(y\right)}^{<\alpha -1>}d{\Gamma }_{\xi ,\eta }(x,y),$$

where ${\left(y\right)}^{<\beta >}={\left|y\right|}^{\beta -1}y$.

For jointly $S\alpha S$ random variables $\xi ={\xi }_1+i{\xi }_2$ and $\eta ={\eta }_1+i{\eta }_2$ the covariation of $\xi$ with $\eta$ is defined as (Cambanis [3])

$${[\xi ,\eta ]}_{\alpha }={\int }_{S_4}(x_1+i{x}_2){(y_1+i{y}_2)}^{<\alpha -1>}d{\Gamma }_{{\xi }_1,{\xi }_2,{\eta }_1,{\eta }_2}(x_1,x_2,y_1,y_2),$$

where $z^{<\beta >}={\left|z\right|}^{\beta -1}\overline{z}$ for a complex number z and $\beta >0.$

The covariation in general is not symmetric and linear on second argument and for $\xi ,{\xi }_1,{\xi }_2,\eta$ jointly $S\alpha S$ has the following properties (Cambanis [3], Weron [27]):

• linearity by the first argument ${[{\xi }_1+{\xi }_2,\eta ]}_{\alpha }={[{\xi }_1,\eta ]}_{\alpha }+{[{\xi }_2,\eta ]}_{\alpha }$,
• linearity by the second argument ${[\xi ,{\eta }_1+{\eta }_2]}_{\alpha }={[\xi ,{\eta }_1]}_{\alpha }+{[\xi ,{\eta }_2]}_{\alpha }$ for the independent ${\eta }_1$ and ${\eta }_2$,
• zero covariation ${[\xi ,\eta ]}_{\alpha }=0$ for the independent $\xi$ and $\eta$,
• ${[a\xi ,b\eta ]}_{\alpha }=a{\left(b\right)}^{\alpha -1}{[\xi ,\eta ]}_{\alpha }$,
• ${|[\xi ,\eta ]}_{\alpha }{|\le \parallel \xi \parallel }_{\alpha }{\parallel \eta \parallel }_{\alpha }^{\alpha -1}$.

The functional

$${\parallel \xi \parallel }_{\alpha }={[\xi ,\xi ]}_{\alpha }^{1/\alpha }$$

(1)

is a norm in a linear space of $S\alpha S$ random variables which is equivalent to convergence in probability. The equality

$$\parallel {\displaystyle \sum _{k=1}^n}{\xi }_k{\parallel }_{\alpha }^{\alpha }={\displaystyle \sum _{k=1}^n}{\parallel {\xi }_k\parallel }_{\alpha }^{\alpha }$$

(2)

holds true for the independent ${\xi }_1,\dots ,{\xi }_n$.

The mapping

$$\xi \to {[\xi ,\eta ]}_{\alpha }$$

(3)

is a bounded linear functional with the norm ${\parallel \eta \parallel }_{\alpha }^{\alpha -1}$ on the linear space of $S\alpha S$ random variables, and every bounded linear functional on such a space is of this form for some $\eta$.

It should be noted that ${\parallel ·\parallel }_{\alpha }$ is not necessarily the usual $L_{\alpha }$ norm.

The following lemma contains the simplest properties of the function $z^{<\beta >}.$

Lemma 1. 

Let $z,x,y$ be complex numbers, $\beta >0$. Then the following properties holds true:

• ${\left|z\right|}^{<\beta >}=z·z^{<\beta -1>},$
• $\left|{\left|z\right|}^{<\beta >}\right|={\left|z\right|}^{<\beta >},$
• if $z^{<\beta >}=v$, then $z=v^{<1/\beta >}={\left|v\right|}^{(1-\beta )/\beta }\overline{v},$
• $z^{<1>}=\overline{z},$
• if $z\ne 0$, then $z^{<\alpha >}{z}^{<\beta >}={\displaystyle \frac{\overline{z}}{\left|z\right|}}{z}^{<\alpha +\beta >},$
• if $z\ne 0$, then ${\displaystyle \frac{z^{<\alpha >}}{z^{<\beta >}}}={\displaystyle \frac{z}{\left|z\right|}}{z}^{<\alpha -\beta >},$
• ${\left(cz\right)}^{<\alpha >}=c^{\alpha }{z}^{<\alpha >},c\in \mathbb{R},$
• ${\left(z^{<\alpha >}\right)}^{<\beta >}={\overline{z}}^{<\alpha \beta >},$
• ${\left(xy\right)}^{<\alpha >}=x^{<\alpha >}{y}^{<\alpha >},$
• ${\left(z^{\alpha }\right)}^{<\beta >}={\left(z^{<\beta >}\right)}^{\alpha },$
• ${\left(z^{<\alpha >}\right)}^{\beta }={\left(z^{\beta }\right)}^{<\alpha >},$
• $|z^{<\alpha >}{|}^{\beta }={\left|z\right|}^{\alpha \beta },$
• ${(x+y)}^{<\alpha >}=\overline{x}{|x+y|}^{\alpha -1}+\overline{y}{|x+y|}^{\alpha -1}.$

Definition 2  

(symmetric $\alpha$-stable stochastic sequence). A stochastic sequence $\left\{\xi \right(n),n\in \mathbb{Z}\}$ is called symmetric α-stable, $S\alpha S$, stochastic sequence, if all linear combinations ${\sum }_{m=1}^l{a}_m\xi \left(n_m\right)$ are $S\alpha S$ random variables.

Let $Z=\left\{Z\right(t):-\infty <t<\infty \}$ be a complex $S\alpha S$ process with independent increments. The spectral measure of the process Z is defined as $\mu \left\{(s,t]\right\}={\parallel Z\left(t\right)-Z\left(s\right)\parallel }_{\alpha }^{\alpha }.$

The integrals $\int a\left(t\right)dZ\left(t\right)$ can be defined for all $a\left(t\right)\in L_{\alpha }\left(\mu \right)$ with properties for all $a\in L_{\alpha }\left(\mu \right),b\in L_{\alpha }\left(\mu \right)$ (see Cambanis [3]; Cambanis and Soltani [4]; Hosoya [8]):

$${∥\int a\left(t\right)dZ\left(t\right)∥}_{\alpha }^{\alpha }=\int {\left|a\left(t\right)\right|}^{\alpha }d\mu \left(t\right),$$

$${\left[\int a\left(t\right)dZ\left(t\right),\int b\left(t\right)dZ\left(t\right)\right]}_{\alpha }=\int a\left(t\right){\left(b\left(t\right)\right)}^{<\alpha -1>}d\mu \left(t\right).$$

Definition 3  

(Harmonizable symmetric $\alpha$-stable sequence). A symmetric α-stable, $S\alpha S$, stochastic sequence $\left\{\xi \right(m),m\in \mathbb{Z}\}$ is said to be harmonizable, $HS\alpha S$, if there exists a $S\alpha S$ process $Z=\left\{Z\right(\lambda ):\lambda \in [-\pi ,\pi \left]\right\}$ with independent increments and finite spectral measure μ such that sequence $\xi \left(m\right)$ has the spectral representation

$$\xi \left(m\right)={\int }_{-\pi }^{\pi }{e}^{im\lambda }dZ\left(\lambda \right),m\in \mathbb{Z},$$

and the covariation has the representation

$${[\xi \left(k\right),\xi \left(m\right)]}_{\alpha }={\int }_{-\pi }^{\pi }{e}^{i(k-m)\lambda }d\mu \left(\lambda \right),k,m\in \mathbb{Z}.$$

1.2. Harmonizable Symmetric $\alpha$-Stable Increment Sequence

Definition 4  

(stochastic nth increment sequence). For a given stochastic sequence $\left\{\xi \right(m),m\in \mathbb{Z}\}$, the sequence

$${\xi }^{\left(n\right)}(m,\mu )={(1-B_{\mu })}^n\xi \left(m\right)={\displaystyle \sum _{l=0}^n}{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right)\xi (m-l\mu ),$$

(4)

where $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right)={\displaystyle \frac{n!}{l!(n-l)!}}$, $B_{\mu }$ denotes a backward shift operator with the step $\mu \in \mathbb{Z}$, such that $B_{\mu }\xi \left(m\right)=\xi (m-\mu )$, is called stochastic nth increment sequence with step $\mu \in \mathbb{Z}$.

Within this paper, assume that $\mu >0$.

Let us give a definition of a harmonizable symmetric $\alpha$-stable increment sequence which is in the scope of interest of this paper.

Definition 5  

(harmonizable symmetric $\alpha$-stable increment sequence). The stochastic nth increment sequence ${\xi }^{\left(n\right)}(m,\mu )$ generated by a symmetric α-stable, $S\alpha S$, stochastic sequence $\left\{\xi \right(m),m\in \mathbb{Z}\}$ is said to be (strongly) harmonizable symmetric α-stable, $HS\alpha S$, if there exists a $S\alpha S$ stochastic process $Z_{{\xi }^{\left(n\right)}}\left(\lambda \right)$ with independent increments on $[-\pi ,\pi )$ and a left-continuous nondecreasing bounded (spectral) function $F\left(\lambda \right)$, $F(-\pi )=0$, such that sequence ${\xi }^{\left(n\right)}(m,\mu )$ has the spectral representation

$${\xi }^{\left(n\right)}(m,\mu )={\int }_{-\pi }^{\pi }{e}^{im\lambda }{(1-e^{-i\mu \lambda })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right),$$

(5)

where the process $Z_{{\xi }^{\left(n\right)}}\left(\lambda \right)$ connected with the spectral function $F\left(\lambda \right)$ by the relation

$$\parallel Z_{{\xi }^{\left(n\right)}}\left(t_2\right)-Z_{{\xi }^{\left(n\right)}}\left(t_1\right){\parallel }_{\alpha }^{\alpha }=F\left(t_2\right)-F\left(t_1\right)<\infty -\pi \le t_1<t_2<\pi .$$

The stochastic sequence $\left\{\xi \right(m),m\in \mathbb{Z}\}$ which determines $HS\alpha S$nth increment sequence ${\xi }^{\left(n\right)}(m,\mu )$ by formula (4) is called stochastic sequence with harmonizable symmetric α-stable nth increments.

Making use of equality and Lemma 1, we obtain a representation of the covariation of a $HS\alpha S$ increment sequence ${\xi }^{\left(n\right)}(m,\mu )$:

$$\begin{array}{c}{[{\xi }^{\left(n\right)}(m_1,{\mu }_1),{\xi }^{\left(n\right)}(m_2,{\mu }_2)]}_{\alpha }\hfill \\ \hfill ={\int }_{-\pi }^{\pi }{e}^{i(m_1-m_2)\lambda }{(1-e^{-i{\mu }_1\lambda })}^n{\left({(1-e^{-i{\mu }_2\lambda })}^n\right)}^{<\alpha -1>}{\displaystyle \frac{1}{{\left|\lambda \right|}^{\alpha n}}}dF\left(\lambda \right),m_1,m_2\in \mathbb{Z}.\end{array}$$

(6)

Note that we will call by spectral function and spectral density of the stochastic sequence with harmonizable increments the spectral function and the spectral density of the corresponding increment sequence.

Consider another stochastic sequence with the $HS\alpha S$ increments $\zeta \left(m\right)=\xi \left(m\right)+\eta \left(m\right)$, where $\left\{\eta \right(m),m\in \mathbb{Z}\}$ is a $HS\alpha S$ stochastic sequence, independent of $\xi \left(m\right)$, with a spectral representation

$$\eta \left(m\right)={\int }_{-\pi }^{\pi }{e}^{i\lambda m}d{Z}_{\eta }\left(\lambda \right),$$

$Z_{\eta }\left(\lambda \right)$, $\lambda \in [-\pi ,\pi )$, is a complex $S\alpha S$ process with uncorrelated increments, that corresponds to the spectral function $G\left(\lambda \right)$. The stochastic $HS\alpha S$ increment sequence ${\zeta }^{\left(n\right)}(m,\mu )$ allows the spectral representation

$${\zeta }^{\left(n\right)}(m,\mu )={\int }_{-\pi }^{\pi }{e}^{i\lambda m}{(1-e^{-i\mu \lambda })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)+{\int }_{-\pi }^{\pi }{e}^{i\lambda m}{(1-e^{-i\mu \lambda })}^{n}d{Z}_{\eta }\left(\lambda \right),$$

where $d{Z}_{\eta }\left(\lambda \right)={\left(i\lambda \right)}^{-n}d{Z}_{{\eta }^{\left(n\right)}}\left(\lambda \right)$, $\lambda \in [-\pi ,\pi )$. If the spectral functions $F\left(\lambda \right)$ and $G\left(\lambda \right)$ have the spectral densities $f\left(\lambda \right)$ and $g\left(\lambda \right)$, the spectral density $p\left(\lambda \right)$ of the stochastic sequence $\zeta \left(m\right)$ is determined by the formula

$$p\left(\lambda \right)=f\left(\lambda \right)+{\left|\lambda \right|}^{\alpha n}g\left(\lambda \right).$$

Let $\left\{\xi \right(m),m\in \mathbb{Z}\}$ and $\left\{\theta \right(m),m\in \mathbb{Z}\}$ be two independent stochastic sequences with $HS\alpha S$nth increments having spectral densities $f\left(\lambda \right)$ and $q\left(\lambda \right)$, respectively. Then $f\left(\lambda \right)+q\left(\lambda \right)$ is a spectral density of the sequence $\left\{\xi \right(m)+\theta (m),m\in \mathbb{Z}\}$.

1.3. Linear Functionals from Sequences with Harmonizable Symmetric $\alpha$-Stable Increments

Consider the functional

$$A_N\xi ={\displaystyle \sum _{k=0}^N}a\left(k\right)\xi \left(k\right).$$

The following Lemma describes its representation in terms of the functionals from the increments of the sequences $\xi \left(m\right)$ or $\zeta \left(m\right)=\xi \left(m\right)+\eta \left(m\right)$.

Lemma 2  

([12]). The functional $A_N\xi$ allows the representations

$$A_N\xi =B_N\xi -V_N\xi$$

(7)

and

$$A_N\overrightarrow{\xi }=A_N\overrightarrow{\zeta }-A_N\overrightarrow{\eta }=H_N\overrightarrow{\xi }-V_N\overrightarrow{\zeta },$$

(8)

where

$$B_N\xi ={\displaystyle \sum _{k=0}^N}{b}_{\mu ,N}\left(k\right){\xi }^{\left(n\right)}(k,\mu )={\displaystyle \sum _{k=0}^N}{\left(D_N^{\mu }{\mathbf{a}}_N\right)}_k{\xi }^{\left(n\right)}(k,\mu ),V_N\xi ={\displaystyle \sum _{k=-\mu n}^{-1}}{v}_{\mu ,N}\left(k\right)\xi \left(k\right)$$

and

$$H_N\xi :=B_N\zeta -A_N\eta ,$$

$$A_N\zeta ={\displaystyle \sum _{k=0}^N}a\left(k\right)\zeta \left(k\right),A_N\eta ={\displaystyle \sum _{k=0}^N}a\left(k\right)\eta \left(k\right),$$

$$B_N\zeta ={\displaystyle \sum _{k=0}^N}{b}_{\mu ,N}\left(k\right){\zeta }^{\left(n\right)}(k,\mu ),V_N\zeta ={\displaystyle \sum _{k=-\mu n}^{-1}}{v}_{\mu ,N}\left(k\right)\zeta \left(k\right).$$

Coefficients $b_{\mu ,N}\left(k\right)$, $k=0,1,2,\dots ,N$, and $v_{\mu ,N}\left(k\right)$, $k=-\mu n,-\mu n+1,\dots ,-1$, are calculated by formulas

$$\begin{array}{cc}\hfill v_{\mu ,N}\left(k\right)& ={\displaystyle \sum _{l=⌈-k/\mu ⌉}^{min\left\{\left[(N-k)/\mu \right],n\right\}}}{(-1)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right)b_{\mu ,N}(l\mu +k),k=-\mu n,-\mu n+1,\dots ,-1,\hfill \\ \hfill b_{\mu ,N}\left(k\right)& ={\displaystyle \sum _{m=k}^N}a\left(m\right)d_{\mu }(m-k)={\left(D_N^{\mu }{\mathbf{a}}_N\right)}_k,k=0,1,\dots ,N,\hfill \end{array}$$

where $\left[x\right]$ is the integer part of x, $\lceil x\rceil$ is the least integer number among those which are larger than or equal to x, coefficients $\{d_{\mu }\left(k\right):k\ge 0\}$ are defined by the relation

$${\displaystyle \sum _{k=0}^{\infty }}{d}_{\mu }\left(k\right)x^k={\left({\displaystyle \sum _{j=0}^{\infty }}{x}^{\mu j}\right)}^n,$$

$D_N^{\mu }$ is the $(N+1)\times (N+1)$ matrix with the entries ${\left(D_N^{\mu }\right)}_{k,j}=d_{\mu }(j-k)$ if $0\le k\le j\le N$, and ${\left(D_N^{\mu }\right)}_{k,j}=0$ if $0\le j<k\le N$; ${\mathbf{a}}_N={(a\left(0\right),a\left(1\right),a\left(2\right),\dots ,a\left(N\right))}^{\top }$ is the vector of dimension $N+1$.

2. Estimation Problem for Stochastic Sequences with Harmonizable Stable nth Increments

2.1. Linear Estimation Problem

Consider the problem of optimal linear estimation of the functional $A_N\xi$ which depends on the unknown values $\xi \left(k\right)$, $k=0,1,\dots ,N$, of a stochastic sequence with harmonizable symmetric $\alpha$-stable nth increments. The linear estimates ${\tilde{A}}_N\xi$ are based on observations $\xi \left(m\right)$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$ (estimation without noise), $\xi \left(m\right)+\eta \left(m\right)$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$ (estimation with noise), or $\xi \left(m\right)$ at points $m\le -1$ and $\xi \left(m\right)+\theta \left(m\right)$ at points $m\ge N+1$ (estimation with semi-noise), where the sequences $\eta \left(m\right)$ and $\theta \left(m\right)$ are defined in SubSection 1.2. To derive the estimates consider the $\alpha$-norm errors, or variances,

$$\begin{array}{cc}\hfill \Delta ({\tilde{A}}_N\xi ;f)& =\parallel A_N\xi -{\tilde{A}}_N{\xi \parallel }_{\alpha }^{\alpha },\hfill \\ \hfill \Delta ({\tilde{A}}_N\xi ;f,g)& =\parallel A_N\xi -{\tilde{A}}_N{\xi \parallel }_{\alpha }^{\alpha },\hfill \\ \hfill \Delta ({\tilde{A}}_N\xi ;f,q)& =\parallel A_N\xi -{\tilde{A}}_N{\xi \parallel }_{\alpha }^{\alpha }\hfill \end{array}$$

for the estimation without noise, with noise or with semi-noise, respectively.

As a classical solution to the estimation problem we consider the optimal linear estimates ${\widehat{A}}_N\xi$ that minimize the values $\Delta ({\tilde{A}}_N\xi ;f)$, $\Delta ({\tilde{A}}_N\xi ;f,g)$ or $\Delta ({\tilde{A}}_N\xi ;f,q)$ for the given spectral densities f, g and q of the sequences $\xi \left(m\right)$, $\eta \left(m\right)$ and $\theta \left(m\right)$.

The functionals $V_N\xi$ and $V_N\zeta$ from representations (7) and (8) depend on the known observations at points $k=-\mu n,-\mu n+1,\dots ,-1$. Thus, the classical solution to the estimation problem can be found in the form

$${\widehat{A}}_N\xi ={\widehat{B}}_N\xi -V_N\xi$$

(9)

or

$${\widehat{A}}_N\xi ={\widehat{H}}_N\xi -V_N\xi ,$$

(10)

where ${\widehat{B}}_N\xi$ is the optimal linear estimate of the functional $B_N\xi$ based on either observations ${\xi }^{\left(n\right)}(m,\mu )$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$ (estimation without noise), or ${\xi }^{\left(n\right)}(m,\mu )$ at points $m\le -1$ and ${\xi }^{\left(n\right)}(m,\mu )+{\theta }^{\left(n\right)}(m,\mu )$ at points $m\ge N+\mu n+1$ (estimation with semi-noise); ${\widehat{H}}_N\xi$ is the optimal linear estimate of the functional $H_N\xi$ based on the observations ${\xi }^{\left(n\right)}(m,\mu )+{\eta }^{\left(n\right)}(m,\mu )$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$ (estimation with noise). Thus, the following relations take place:

$$\begin{array}{cc}\hfill \Delta ({\widehat{A}}_N\xi ;f)& =\Delta ({\widehat{B}}_N\xi ;f)={\parallel B_N\xi -{\widehat{B}}_N\xi \parallel }_{\alpha }^{\alpha },\hfill \\ \hfill \Delta ({\widehat{A}}_N\xi ;f,g)& =\Delta ({\widehat{H}}_N\xi ;f,g)={\parallel H_N\xi -{\widehat{H}}_N\xi \parallel }_{\alpha }^{\alpha },\hfill \\ \hfill \Delta ({\widehat{A}}_N\xi ;f,q)& =\Delta ({\widehat{B}}_N\xi ;f,q)={\parallel B_N\xi -{\widehat{B}}_N\xi \parallel }_{\alpha }^{\alpha }.\hfill \end{array}$$

Denote by $H^{0-}\left({\xi }_{\mu }^{\left(n\right)}\right)$ the closed in the ${\parallel ·\parallel }_{\alpha }$ norm linear manifold generated by the values $\{{\xi }^{\left(n\right)}(k,\mu ):k\le -1\}$ of the harmonizable symmetric $\alpha$-stable random sequence in the space H generated by all values of $\{{\xi }^{\left(n\right)}(k,\mu ):k\in \mathbb{Z}\}$, and denote by $H^{N+}\left({\xi }_{-\mu }^{\left(n\right)}\right)$ the closed linear manifold generated by values $\{{\xi }^{\left(n\right)}(k,-\mu ):k\ge N+1\}$. Note that

$$H^{N+}\left({\xi }_{-\mu }^{\left(n\right)}\right)=H^{(N+\mu n)+}\left({\xi }_{\mu }^{\left(n\right)}\right).$$

In the same way as above, define the closed in the ${\parallel ·\parallel }_{\alpha }$ norm linear manifolds $H^{0-}\left({\zeta }_{\mu }^{\left(n\right)}\right)$ and $H^{(N+\mu n)+}\left({\zeta }_{\mu }^{\left(n\right)}\right)$.

Define also the subspace $L_{\alpha }^{0-}\left(s\right)\oplus L_{\alpha }^{(N+\mu n)+}\left(s\right)$ in the space $L_{\alpha }\left(s\right)$ generated by the functions

$$\left\{e^{i\lambda k}{(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}:k\le -1\right\}\bigcup \left\{e^{i\lambda k}{(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}:k\ge N+\mu n+1\right\},$$

where the functions $f\left(\lambda \right)$, $p\left(\lambda \right)=f\left(\lambda \right)+{\left|\lambda \right|}^{\alpha n}g\left(\lambda \right)$, $\left(f\right(\lambda ),f(\lambda )+q(\lambda \left)\right)$ can be chosen as $s\left(\lambda \right)$ in case of estimation without noise, with noise or with semi-noise, respectively.

There is a map between the elements $e^{i\lambda k}{(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}$ of the space $L_{\alpha }\left(f\right)$ and the elements ${\xi }^{\left(n\right)}(k,\mu )$ of the space H, and a map between the elements $e^{i\lambda k}{(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}$ of the space $L_{\alpha }{(f+|\lambda |}^{\alpha n}g)$ and the elements ${\xi }^{\left(n\right)}(k,\mu )+{\eta }^{\left(n\right)}(k,\mu )$ of the space H.

The optimal linear estimate can be found as a projection of the target functional on the manifold generated by the observations [26].

The optimal linear estimate ${\widehat{B}}_N\xi$ is characterised by the conditions

A1: ${[{\xi }^{\left(n\right)}(k,\mu ),B_N\xi -{\widehat{B}}_N\xi ]}_{\alpha }=0$, $k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$;

A2: ${\widehat{B}}_N\xi \in H^{0-}\left({\xi }_{\mu }^{\left(n\right)}\right)\oplus H^{(N+\mu n)+}\left({\xi }_{\mu }^{\left(n\right)}\right)$.

The optimal linear estimate ${\widehat{H}}_N\xi$ is characterised by the conditions

B1: ${[{\xi }^{\left(n\right)}(k,\mu )+{\eta }^{\left(n\right)}(k,\mu ),H_N\xi -{\widehat{H}}_N\xi ]}_{\alpha }=0$, $k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$;

B2: ${\widehat{H}}_N\xi \in H^{0-}({\xi }_{\mu }^{\left(n\right)}+{\eta }_{\mu }^{\left(n\right)})\oplus H^{(N+\mu n)+}({\xi }_{\mu }^{\left(n\right)}+{\eta }_{\mu }^{\left(n\right)})$.

The optimal linear estimate ${\widehat{B}}_N\xi$ is characterised by the conditions

C1.1: ${[{\xi }^{\left(n\right)}(k,\mu ),B_N\xi -{\widehat{B}}_N\xi ]}_{\alpha }=0$, $k\le -1$;

C1.2: ${[{\xi }^{\left(n\right)}(k,\mu )+{\theta }^{\left(n\right)}(k,\mu ),B_N\xi -{\widehat{B}}_N\xi ]}_{\alpha }=0$, $k\ge N+\mu n+1$;

C2: ${\widehat{B}}_N\xi \in H^{0-}\left({\xi }_{\mu }^{\left(n\right)}\right)\oplus H^{(N+\mu n)+}({\xi }_{\mu }^{\left(n\right)}+{\xi }_{\mu }^{\left(n\right)})$.

For further analysis, we consider the following minimality conditions for the spectral densities $f\left(\lambda \right)$, $q\left(\lambda \right)$ and $g\left(\lambda \right)$ [22,27]

$${\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}f\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}d\lambda <\infty ,{\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}q\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}d\lambda <\infty ,$$

(11)

$${\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}{(f\left(\lambda \right)+|\lambda |}^{\alpha n}g\left(\lambda \right))}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}d\lambda <\infty .$$

(12)

As a minimax solution to the estimation problem we consider the estimates ${\widehat{A}}_N\xi$ which minimize the maximal values of $\Delta ({\tilde{A}}_N\xi ;f)$, $\Delta ({\tilde{A}}_N\xi ;f,g)$ or $\Delta ({\tilde{A}}_N\xi ;f,q)$ for all spectral densities f, g and q from the given classes $\mathcal{D}={\mathcal{D}}_f$, $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_g$ and $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_q$ of admissible spectral densities simultaneously. Taking into account the maps between the subspaces of the spaces H and $L_{\alpha }\left(s\right)$ stated above, this approach is formalized by the following definitions.

Definition 6.  

For the given classes of spectral densities $\mathcal{D}={\mathcal{D}}_f$, $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_g$ and $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_q$, the spectral densities $f^0\left(\lambda \right)\in {\mathcal{D}}_f$, $g^0\left(\lambda \right)\in {\mathcal{D}}_g$, $q^0\left(\lambda \right)\in {\mathcal{D}}_q$ are called least favorable in the classes $\mathcal{D}$ for the optimal linear estimation of the functional $A_N\xi$ if the following relations hold true:

$$\begin{array}{cc}\hfill \Delta \left(f^0\right)& =\Delta (h\left(f^0\right);f^0)=\underset{f\in {\mathcal{D}}_f}{max}\Delta (h\left(f\right);f),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \Delta (f^0,g^0)& =\Delta (h(f^0,g^0);f^0,g^0)=\underset{(f,g)\in {\mathcal{D}}_f\times {\mathcal{D}}_g}{max}\Delta (h(f,g);f,g),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \Delta (f^0,q^0)& =\Delta (h(f^0,q^0);f^0,q^0)=\underset{(f,q)\in {\mathcal{D}}_f\times {\mathcal{D}}_q}{max}\Delta (h(f,q);f,q),\hfill \end{array}$$

(15)

where $h\left(f^0\right)$, $h(f^0,g^0)$, $h(f^0,q^0)$ are the spectral characteristics of the optimal estimates ${\tilde{A}}_N\xi$ for the estimation without noise, estimation with noise, estimation with semi-noise problems, respectively.

Definition 7.  

For the given classes of spectral densities $\mathcal{D}={\mathcal{D}}_f$, $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_g$ and $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_q$, the spectral characteristics $h^0\left(\lambda \right)$ of the optimal linear estimates of the functional $A_N\xi$ are called minimax-robust if there are satisfied the conditions:

for the estimationn without noise:

$$h^0\left(\lambda \right)\in H_{\mathcal{D}}=\bigcap _{f\in {\mathcal{D}}_f}{L}_{\alpha }^{0-}\left(f\right)\oplus L_{\alpha }^{(N+\mu n)+}\left(f\right),$$

$$\underset{h\in H_{\mathcal{D}}}{min}\underset{f\in {\mathcal{D}}_f}{max}\Delta (h;f)=\underset{f\in {\mathcal{D}}_f}{max}\Delta (h^0;f);$$

for the estimation with noise:

$$h^0\left(\lambda \right)\in H_{\mathcal{D}}=\bigcap _{(f,g)\in {\mathcal{D}}_f\times {\mathcal{D}}_g}{L}_{\alpha }^{0-}{(f+|\lambda |}^{\alpha n}g)\oplus L_{\alpha }^{(N+\mu n)+}{(f+|\lambda |}^{\alpha n}g),$$

$$\underset{h\in H_{\mathcal{D}}}{min}\underset{(f,g)\in {\mathcal{D}}_f\times {\mathcal{D}}_g}{max}\Delta (h;f,g)=\underset{(f,g)\in {\mathcal{D}}_f\times {\mathcal{D}}_g}{max}\Delta (h^0;f,g);$$

for the estimation with semi-noise:

$$h^0\left(\lambda \right)\in H_{\mathcal{D}}=\bigcap _{(f,q)\in {\mathcal{D}}_f\times {\mathcal{D}}_q}{L}_{\alpha }^{0-}\left(f\right)\oplus L_{\alpha }^{(N+\mu n)+}(f+q),$$

$$\underset{h\in H_{\mathcal{D}}}{min}\underset{(f,q)\in {\mathcal{D}}_f\times {\mathcal{D}}_q}{max}\Delta (h;f,q)=\underset{(f,q)\in {\mathcal{D}}_f\times {\mathcal{D}}_q}{max}\Delta (h^0;f,q).$$

In the next section, we provide the solutions to the classical and minimax estimation problems.

3. Main Results

3.1. Solution to Classical Estimation Problem Without Noise: Projection Method of Estimation

Theorem 1.  

Let a stochastic sequence $\left\{\xi \right(m),m\in \mathbb{Z}\}$ generate the $HS\alpha S$nth increment sequence $\{{\xi }^{\left(n\right)}(m,\mu ),m\in \mathbb{Z}\}$ with the absolutely continuous spectral function $F\left(\lambda \right)$ which has the spectral density $f\left(\lambda \right)$ satisfying minimality condition (11). The optimal linear estimate ${\widehat{A}}_N\xi$ of the unknown value of functional $A_N\xi$, from observations $\xi \left(m\right)$ at points of the set $\mathbb{Z}\setminus \{0,1,2,\dots ,N\}$ is calculated by the formula

$${\widehat{A}}_N\xi ={\int }_{-\pi }^{\pi }{h}_{\mu }\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)-{\displaystyle \sum _{k=-\mu n}^{-1}}{v}_{\mu ,N}\left(k\right)\xi \left(k\right).$$

(16)

The spectral characteristic $h_{\mu }\left(\lambda \right)$ of the estimate ${\widehat{A}}_N\xi$ has the form

$$h_{\mu }\left(\lambda \right)=B_N^{\mu }\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}-{\left({\displaystyle \frac{{\left(i\lambda \right)}^n{C}_{\mu ,N}\left(e^{i\lambda }\right)}{{(1-e^{-i\lambda \mu })}^{n}f\left(\lambda \right)}}\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>},$$

(17)

where

$$B_N^{\mu }\left(e^{i\lambda }\right)={\displaystyle \sum _{k=0}^N}{\left(D_N^{\mu }{\mathbf{a}}_N\right)}_k{e}^{i\lambda k},C_{\mu ,N}\left(e^{i\lambda }\right)={\displaystyle \sum _{k=0}^{N+\mu n}}{c}_{\mu }\left(k\right)e^{-i\lambda k},$$

and the coefficients $c_{\mu }\left(k\right)$, $k=0,1,2,\dots ,N+\mu n$ are determined from the system of equations

$$\begin{array}{c}{\int }_{-\pi }^{\pi }\left[B_N^{\mu }\left(e^{i\lambda }\right)-{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}f\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{C}_{\mu ,N}{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right]e^{-i\lambda l}d\lambda =0,\hfill \\ \hfill l=0,1,\dots ,N+\mu n.\end{array}$$

(18)

The variance of the estimate ${\widehat{A}}_N\xi$ is calculated by the formula

$$\Delta ({\widehat{A}}_N\xi ;f)={\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{\left|{\left({\displaystyle \frac{{\left(i\lambda \right)}^n{C}_{\mu ,N}\left(e^{i\lambda }\right)}{{(1-e^{-i\lambda \mu })}^{n}f\left(\lambda \right)}}\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }f\left(\lambda \right)d\lambda .$$

(19)

Proof. 

Equality (9) implies representation(16) of the optimal estimate ${\widehat{A}}_N\xi$, where the $h_{\mu }\left(\lambda \right)$ is a spectral characteristic of the estimate ${\widehat{B}}_N\xi$.

Making use of Condition A1 and the spectral representation

$$B_N\xi ={\int }_{-\pi }^{\pi }{B}_N^{\mu }\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right),B_N^{\mu }\left(e^{i\lambda }\right)={\displaystyle \sum _{k=0}^N}{\left(D_N^{\mu }{\mathbf{a}}_N\right)}_k{e}^{i\lambda k},$$

(20)

obtain the following equations for all $l\le -1$ and $l\ge N+\mu n+1$:

$${\int }_{-\pi }^{\pi }{e}^{i\lambda l}{(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}f\left(\lambda \right){\left(B_N^{\mu }\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}-h_{\mu }\left(\lambda \right)\right)}^{<\alpha -1>}d\lambda =0,$$

from which one can derive representation (17) of the spectral characteristic $h_{\mu }\left(\lambda \right)$ with the unknown coefficients $c_{\mu }\left(k\right)$, $k=0,1,2,\dots ,N+\mu n$.

Making use of Condition A2 conclude that the spectral characteristic $h_{\mu }\left(\lambda \right)$ has the form

$$h_{\mu }\left(\lambda \right)=H_{\mu }\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}},$$

where

$$H_{\mu }\left(e^{i\lambda }\right)={\displaystyle \sum _{k=-\infty }^{-1}}{h}_{\mu }\left(k\right)e^{i\lambda k}+{\displaystyle \sum _{k=N+\mu n+1}^{\infty }}{h}_{\mu }\left(k\right)e^{i\lambda k},$$

which allows us to derive the following system of equations determining the coefficients $c_{\mu }\left(k\right)$, $k=0,1,2,\dots ,N+\mu n$:

$$\begin{array}{c}{\int }_{-\pi }^{\pi }\left[B_N^{\mu }\left(e^{i\lambda }\right)-{\displaystyle \frac{{\left(i\lambda \right)}^n}{{(1-e^{-i\lambda \mu })}^n}}{\left({\displaystyle \frac{{\left(i\lambda \right)}^n{C}_{\mu ,N}\left(e^{i\lambda }\right)}{{(1-e^{-i\lambda \mu })}^{n}f\left(\lambda \right)}}\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right]e^{-i\lambda l}d\lambda =0,\hfill \\ \hfill l=0,1,\dots ,N+\mu n.\end{array}$$

Thus, making use of Lemma 1 one can obtain the system (18) from the statement of the theorem.

Expression (20) for the spectral representation of the functional $B_N\xi$ and the derived expressions for the optimal estimate ${\widehat{A}}_N\xi$ justify formula (19) for the variance $\Delta ({\widehat{A}}_N\xi ;f)$. □

3.2. Solution to Classical Estimation Problem with Noise: Projection Method of Estimation

Theorem 2.  

Let a stochastic sequence $\left\{\xi \right(m),m\in \mathbb{Z}\}$ generate the $HS\alpha S$nth increment sequence $\{{\xi }^{\left(n\right)}(m,\mu ),m\in \mathbb{Z}\}$ with the absolutely continuous spectral function $F\left(\lambda \right)$ which has the spectral density $f\left(\lambda \right)$. Let $\left\{\eta \right(m),m\in \mathbb{Z}\}$ be an independent from $\left\{\xi \right(m),m\in \mathbb{Z}\}$$HS\alpha S$ stochastic sequence with the absolutely continuous spectral function $G\left(\lambda \right)$ which has the spectral density $g\left(\lambda \right)$. Assume that $f\left(\lambda \right)$ and $g\left(\lambda \right)$ satisfy minimality condition (11). The optimal linear estimate ${\widehat{A}}_N\xi$ of the unknown value of functional $A_N\xi$, from observations $\xi \left(m\right)+\eta \left(m\right)$ at points of the set $\mathbb{Z}\setminus \{0,1,2,\dots ,N\}$ is calculated by the formula

$${\widehat{A}}_N\xi ={\int }_{-\pi }^{\pi }{h}_{\mu }\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)-{\displaystyle \sum _{k=-\mu n}^{-1}}{v}_{\mu ,N}\left(k\right)(\xi \left(k\right)+\eta \left(k\right)).$$

(21)

The spectral characteristic $h_{\mu }\left(\lambda \right)$ of the optimal estimate has the form

$$h_{\mu }\left(\lambda \right)=H_{\mu }\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}},$$

(22)

where

$$H_{\mu }\left(e^{i\lambda }\right)={\displaystyle \sum _{k=-\infty }^{-1}}{h}_{\mu }\left(k\right)e^{i\lambda k}+{\displaystyle \sum _{k=N+\mu n+1}^{\infty }}{h}_{\mu }\left(k\right)e^{i\lambda k}.$$

The coefficients $h_{\mu }\left(k\right)$, $k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$ are determined from the system of equations

$$\begin{array}{c}{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\left(B_N^{\mu }\left(e^{i\lambda }\right)-H_{\mu }\left(e^{i\lambda }\right)\right)}^{<\alpha -1>}{\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda \hfill \\ +{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\left(B_N^{\mu }\left(e^{i\lambda }\right)-A_N\left(e^{i\lambda }\right){\displaystyle \frac{1}{{(1-e^{-i\lambda \mu })}^n}}-H_{\mu }\left(e^{i\lambda }\right)\right)}^{<\alpha -1>}{|1-e^{-i\lambda \mu }|}^{\alpha n}g\left(\lambda \right)d\lambda =0,\\ \hfill l\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\},\end{array}$$

(23)

where $B_N^{\mu }\left(e^{i\lambda }\right)$ is defined in Theorem 1,

$$A_N\left(e^{i\lambda }\right)={\displaystyle \sum _{k=0}^N}a\left(k\right)e^{i\lambda k}.$$

The variance of the optimal estimate ${\widehat{A}}_N\xi$ is calculated by the formula

$$\begin{array}{cc}\hfill \Delta ({\widehat{A}}_N\xi ;f,g)& ={\int }_{-\pi }^{\pi }{\left|B_N^{\mu }\left(e^{i\lambda }\right)-H_{\mu }\left(e^{i\lambda }\right)\right|}^{\alpha }{\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda \hfill \\ & +{\int }_{-\pi }^{\pi }{\left|B_N^{\mu }\left(e^{i\lambda }\right)-A_N\left(e^{i\lambda }\right){\displaystyle \frac{1}{{(1-e^{-i\lambda \mu })}^n}}-H_{\mu }\left(e^{i\lambda }\right)\right|}^{\alpha }{|1-e^{-i\lambda \mu }|}^{\alpha n}g\left(\lambda \right)d\lambda .\hfill \end{array}$$

(24)

Proof. 

Condition B2 implies representation (22) of the spectral characteristic $h_{\mu }\left(\lambda \right)$ with unknown coefficients $h_{\mu }\left(k\right)$, $k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$, to be found.

Equality (10) implies representation(21) of the optimal estimate ${\widehat{A}}_N\xi$, where $h_{\mu }\left(\lambda \right)$ is a spectral characteristic of the estimate ${\widehat{H}}_N\xi$. The functional $H_N\xi =B_N\zeta -A_N\eta$ admits the spectral representation

$$H_N\xi ={\int }_{-\pi }^{\pi }{B}_N^{\mu }\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}d{Z}_{{\xi }^{\left(n\right)}+{\eta }^{\left(n\right)}}\left(\lambda \right)-{\int }_{-\pi }^{\pi }{A}_N\left(e^{i\lambda }\right)d{Z}_{\eta }\left(\lambda \right).$$

(25)

Thus,

$$\begin{array}{cc}\hfill H_N\xi -{\widehat{H}}_N\xi & ={\int }_{-\pi }^{\pi }\left(B_N^{\mu }\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}-h_{\mu }\left(\lambda \right)\right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)\hfill \\ & +{\int }_{-\pi }^{\pi }\left(B_N^{\mu }\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}-A_N\left(e^{i\lambda }\right){\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}-h_{\mu }\left(\lambda \right)\right)d{Z}_{{\eta }^{\left(n\right)}}\left(\lambda \right)\hfill \\ & =:{\int }_{-\pi }^{\pi }{U}_{\xi }^{\mu }\left(e^{i\lambda }\right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)+{\int }_{-\pi }^{\pi }{U}_{\eta }^{\mu }\left(e^{i\lambda }\right)d{Z}_{{\eta }^{\left(n\right)}}\left(\lambda \right).\hfill \end{array}$$

Recall some covariation properties from Lemma 1, namely linearity by the first argument, linearity by the second argument for independent random variables and zero covariation value for the independent random variables. Then making use of the independence of the sequences $\left\{\xi \right(m\left)\right\}$ and $\left\{\eta \right(m\left)\right\}$, as well as Condition B1, obtain the following equations for all $l\le -1$ and $l\ge N+\mu n+1$:

$$\begin{array}{cc}\hfill & {\left[{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right);{\int }_{-\pi }^{\pi }{U}_{\xi }^{\mu }\left(e^{i\lambda }\right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)\right]}_{\alpha }\hfill \\ & +{\left[{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}d{Z}_{{\eta }^{\left(n\right)}}\left(\lambda \right);{\int }_{-\pi }^{\pi }{U}_{\eta }^{\mu }\left(e^{i\lambda }\right)d{Z}_{{\eta }^{\left(n\right)}}\left(\lambda \right)\right]}_{\alpha }\hfill \\ & ={\int }_{-\pi }^{\pi }{e}^{i\lambda l}{(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}{\left(U_{\xi }^{\mu }\left(e^{i\lambda }\right)\right)}^{<\alpha -1>}f\left(\lambda \right)d\lambda \hfill \\ & +{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}{\left(U_{\eta }^{\mu }\left(e^{i\lambda }\right)\right)}^{<\alpha -1>}{\left|\lambda \right|}^{\alpha n}g\left(\lambda \right)d\lambda \hfill \\ & =0.\hfill \end{array}$$

These equations for the unknown coefficients $h_{\mu }\left(k\right)$, $k\in \mathbb{Z}\setminus \{0,1,2,\dots ,N+\mu n\}$, of the spectral characteristic $h_{\mu }\left(\lambda \right)$ are rewritten in the form of system (23) from the theorem statement.

Expression (25) for the spectral representation of the functional $H_N\xi$ and the derived expressions for the optimal estimate ${\widehat{A}}_N\xi$ justify formula (24) for the variance $\Delta ({\widehat{A}}_N\xi ;f,g)$. □

3.3. Solution to Classical Estimation Problem with Semi-Noise: Projection Method of Estimation

Theorem 3.  

Let the stochastic sequences $\left\{\xi \right(m),m\in \mathbb{Z}\}$ and $\left\{\theta \right(m),m\in \mathbb{Z}\}$ generate the $HS\alpha S$nth increment sequences $\{{\xi }^{\left(n\right)}(m,\mu ),m\in \mathbb{Z}\}$ and $\{{\theta }^{\left(n\right)}(m,\mu ),m\in \mathbb{Z}\}$ with the absolutely continuous spectral functions $F\left(\lambda \right)$ and $Q\left(\lambda \right)$ which have the spectral densities $f\left(\lambda \right)$ and $q\left(\lambda \right)$ satisfying minimality condition (11). The optimal linear estimate ${\widehat{A}}_N\xi$ of the unknown value of functional $A_N\xi$, from observations $\xi \left(m\right)$ at points $m\le -1$ and $\xi \left(m\right)+\theta \left(m\right)$ at points $m\ge N+1$ is calculated by the formula

$${\widehat{A}}_N\xi ={\int }_{-\pi }^{\pi }{h}_{\mu }^1\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)+{\int }_{-\pi }^{\pi }{h}_{\mu }^2\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}+{\theta }^{\left(n\right)}}\left(\lambda \right)-{\displaystyle \sum _{k=-\mu n}^{-1}}{v}_{\mu ,N}\left(k\right)\xi \left(k\right).$$

(26)

The spectral characteristic $h_{\mu }\left(\lambda \right)=(h_{\mu }^1\left(\lambda \right),h_{\mu }^2\left(\lambda \right))$ of the estimate ${\widehat{A}}_N\xi$ has the form

$$\begin{array}{cc}\hfill h_{\mu }^1\left(\lambda \right)& =B_N^{\mu }\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}-U_{\mu ,N}^f{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}-U_{\mu ,N}^q{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill h_{\mu }^2\left(\lambda \right)& =U_{\mu ,N}^q{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>},\hfill \end{array}$$

(28)

where $B_N^{\mu }\left(e^{i\lambda }\right)$ is defined in Theorem 1,

$$U_{\mu ,N}^f\left(e^{i\lambda }\right)={\displaystyle \frac{{\left(i\lambda \right)}^n{C}_{\mu ,N}^1\left(e^{i\lambda }\right)}{{(1-e^{-i\lambda \mu })}^{n}f\left(\lambda \right)}},U_{\mu ,N}^q\left(e^{i\lambda }\right)={\displaystyle \frac{{\left(i\lambda \right)}^n(C_{\mu ,N}^1\left(e^{i\lambda }\right)-C_{\mu ,N}^2\left(e^{i\lambda }\right))}{{(1-e^{-i\lambda \mu })}^{n}q\left(\lambda \right)}},$$

and

$$C_{\mu ,N}^1\left(e^{i\lambda }\right)={\displaystyle \sum _{k=0}^{\infty }}{c}_{\mu }^1\left(k\right)e^{-i\lambda k},C_{\mu ,N}^2\left(e^{i\lambda }\right)={\displaystyle \sum _{k=-\infty }^{N+\mu n}}{c}_{\mu }^2\left(k\right)e^{-i\lambda k}.$$

The coefficients $c_{\mu }^1\left(k\right)$, $k\ge 0$, and $c_{\mu }^2\left(k\right)$, $k\le N+\mu n$, are determined from the systems of equations

$$\begin{array}{c}{\int }_{-\pi }^{\pi }\left[B_N^{\mu }\left(e^{i\lambda }\right)-{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}f\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{C}_{\mu ,N}^1{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right.\hfill \\ \left.-{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}q\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{(C_{\mu ,N}^1\left(e^{i\lambda }\right)-C_{\mu ,N}^2\left(e^{i\lambda }\right))}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right]e^{-i\lambda l}d\lambda =0,\\ \hfill l\ge 0,\end{array}$$

(29)

and

$$\begin{array}{c}{\int }_{-\pi }^{\pi }\left[{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}q\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{(C_{\mu ,N}^1\left(e^{i\lambda }\right)-C_{\mu ,N}^2\left(e^{i\lambda }\right))}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right]e^{-i\lambda l}d\lambda =0,\hfill \\ \hfill l\le N+\mu n.\end{array}$$

(30)

The variance of the optimal estimate ${\widehat{A}}_N\xi$ is calculated by the formula

$$\Delta ({\widehat{A}}_N\xi ;f,q)={\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{\left|U_{\mu ,N}^f{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }f\left(\lambda \right)d\lambda +{\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{\left|U_{\mu ,N}^q{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }q\left(\lambda \right)d\lambda .$$

(31)

Proof. 

Let us consider the spectral representation of ${\widehat{B}}_N\xi$ in the form

$${\widehat{B}}_N\xi ={\int }_{-\pi }^{\pi }{h}_{\mu }^1\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)+{\int }_{-\pi }^{\pi }{h}_{\mu }^2\left(\lambda \right)d{Z}_{{\xi }^{\left(n\right)}+{\theta }^{\left(n\right)}}\left(\lambda \right).$$

Then equality (9) implies representation(26) of the optimal estimate ${\widehat{A}}_N\xi$.

Making use of spectral representation (20) of the functional $B_N\xi$ obtain

$$\begin{array}{cc}\hfill B_N\xi -{\widehat{B}}_N\xi & ={\int }_{-\pi }^{\pi }\left(B_N^{\mu }\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}-h_{\mu }^1\left(\lambda \right)-h_{\mu }^2\left(\lambda \right)\right)d{Z}_{{\xi }^{\left(n\right)}}\left(\lambda \right)\hfill \\ & -{\int }_{-\pi }^{\pi }{h}_{\mu }^2\left(\lambda \right)d{Z}_{{\theta }^{\left(n\right)}}\left(\lambda \right).\hfill \end{array}$$

Lemma 1 states the covariation linearity by the second argument for independent random variables and zero covariation value for the independent random variables. Thus, making use of the independence of the sequences $\left\{\xi \right(m\left)\right\}$ and $\left\{\theta \right(m\left)\right\}$, as well as Condition C1.1, we obtain the following equations for all $l\le -1$

$${\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}f\left(\lambda \right){\left(B_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}-h_{\mu }^1\left(\lambda \right)-h_{\mu }^2\left(\lambda \right)\right)}^{<\alpha -1>}d\lambda =0,$$

from which we derive the representation

$$h_{\mu }^1\left(\lambda \right)+h_{\mu }^2\left(\lambda \right)=B_N^{\mu }\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}}-{\left({\displaystyle \frac{{\left(i\lambda \right)}^n{C}_{\mu ,N}^1\left(e^{i\lambda }\right)}{{(1-e^{-i\lambda \mu })}^{n}f\left(\lambda \right)}}\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}$$

(32)

with the unknown coefficients $c_{\mu }^1\left(k\right)$, $k\ge 0$, defining $C_{\mu ,N}^1\left(e^{i\lambda }\right)$ from the theorem statement.

Making use of the covariation linearity by the first argument, covariation properties stated above and Condition C1.2 obtain the following equations for all $l\ge N+\mu n+1$

$$\begin{array}{c}{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}f\left(\lambda \right){\left(B_N^{\mu }\left(e^{i\lambda }\right){\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}-h_{\mu }^1\left(\lambda \right)-h_{\mu }^2\left(\lambda \right)\right)}^{<\alpha -1>}\hfill \\ \hfill -{\int }_{-\pi }^{\pi }{e}^{i\lambda l}{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}g\left(\lambda \right){\left(h_{\mu }^2\left(\lambda \right)\right)}^{<\alpha -1>}d\lambda =0,\end{array}$$

from which we derive the representation

$$C_{\mu ,N}^1\left(e^{i\lambda }\right)-{\displaystyle \frac{{(1-e^{-i\lambda \mu })}^n}{{\left(i\lambda \right)}^n}}q\left(\lambda \right){\left(h_{\mu }^2\left(\lambda \right)\right)}^{<\alpha -1>}=C_{\mu ,N}^2\left(e^{i\lambda }\right)$$

with the unknown coefficients $c_{\mu }^2\left(k\right)$, $k\le N+\mu n$, defining $C_{\mu ,N}^2\left(e^{i\lambda }\right)$ from the theorem statement. Thus,

$$h_{\mu }^2\left(\lambda \right)={\left({\displaystyle \frac{{\left(i\lambda \right)}^n(C_{\mu ,N}^1\left(e^{i\lambda }\right)-C_{\mu ,N}^2\left(e^{i\lambda }\right))}{{(1-e^{-i\lambda \mu })}^{n}q\left(\lambda \right)}}\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}.$$

(33)

Representations (32) and (33) imply formulas (27) and (27) for the spectral characteristic $h_{\mu }\left(\lambda \right)=(h_{\mu }^1\left(\lambda \right),h_{\mu }^2\left(\lambda \right))$ of the estimate ${\widehat{A}}_N\xi$.

Making use of Condition C2 conclude that the spectral characteristic $h_{\mu }\left(\lambda \right)=(h_{\mu }^1\left(\lambda \right),h_{\mu }^2\left(\lambda \right))$ has the form

$$\begin{array}{cc}\hfill h_{\mu }^1\left(\lambda \right)& =H_{\mu }^1\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}},H_{\mu }^1\left(e^{i\lambda }\right)={\displaystyle \sum _{k=-\infty }^{-1}}{h}_{\mu }^1\left(k\right)e^{i\lambda k},\hfill \\ \hfill h_{\mu }^2\left(\lambda \right)& =H_{\mu }^2\left(e^{i\lambda }\right){(1-e^{-i\lambda \mu })}^n{\displaystyle \frac{1}{{\left(i\lambda \right)}^n}},H_{\mu }^2\left(e^{i\lambda }\right)={\displaystyle \sum _{k=N+\mu n+1}^{\infty }}{h}_{\mu }^2\left(k\right)e^{i\lambda k},\hfill \end{array}$$

which allows us to derive the following system of equations determining the coefficients $c_{\mu }^1\left(k\right)$, $k\ge 0$, and $c_{\mu }^2\left(k\right)$, $k\le N+\mu n$, respectively:

$$\begin{array}{cc}\hfill {\int }_{-\pi }^{\pi }{h}_{\mu }^1\left(\lambda \right){\displaystyle \frac{{\left(i\lambda \right)}^n}{{(1-e^{-i\lambda \mu })}^n}}{e}^{-i\lambda l}d\lambda & =0,l\ge 0,\hfill \\ \hfill {\int }_{-\pi }^{\pi }{h}_{\mu }^2\left(\lambda \right){\displaystyle \frac{{\left(i\lambda \right)}^n}{{(1-e^{-i\lambda \mu })}^n}}{e}^{-i\lambda l}d\lambda & =0,l\le N+\mu n.\hfill \end{array}$$

Thus, making use of Lemma 1 one can obtain the system of equations (29)-(30) from the theorem statement.

Expression (20) for the spectral representation of the functional $B_N\xi$, the derived expressions for the optimal estimate ${\widehat{A}}_N\xi$ as well as covariation properties stated above let us justify formula (31) for the variance $\Delta ({\widehat{A}}_N\xi ;f,q)$. □

3.4. Solution to Minimax Estimation Problems

Taking into account the definitions of the minimax estimation approach and the derived solutions to the classical estimation problems we can conclude that the following lemma holds true.

Lemma 3.  

The spectral densities $f^0\in {\mathcal{D}}_f$, $g^0\in {\mathcal{D}}_g$, $q^0\in {\mathcal{D}}_q$ which satisfy minimality conditions (11), (12) are least favorable in the classes $\mathcal{D}={\mathcal{D}}_f$, $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_g$, $\mathcal{D}={\mathcal{D}}_f\times {\mathcal{D}}_q$ for the optimal linear estimation of the functional $A_N\overrightarrow{\xi }$ for the estimation without noise, estimation with noise, estimation with semi-noise problems, respectively, if the functions $f^0\left(\lambda \right)$, $g^0\left(\lambda \right)$, $q^0\left(\lambda \right)$ determine solutions of constraint optimisation problems (13), () and (), where $\Delta \left(h\right(f);f)$, $\Delta \left(h\right(f,g);f,g)$ and $\Delta \left(h\right(f,q);f,q)$ are defined by formulas (19), (24) and (31), respectively. The minimax spectral characteristics $h^0=h_{\mu }\left(f^0\right)$, $h^0=h_{\mu }(f^0,g^0)$ and $h^0=h_{\mu }(f^0,q^0)$ are calculated by formulas (17), (22) and (27)-() if $h_{\mu }\left(f^0\right)\in H_{{\mathcal{D}}_f}$, $h_{\mu }(f^0,g^0)\in H_{{\mathcal{D}}_f\times {\mathcal{D}}_g}$ and $h_{\mu }(f^0,q^0)\in H_{{\mathcal{D}}_f\times {\mathcal{D}}_q}$, respectively.

The minimax spectral characteristic $h^0$ and the least favourable spectral densities $f^0$, $(f^0,g^0)$ or $(f^0,q^0)$ form the saddle points of the functions $\Delta (h;f)$, $\Delta (h;f,g)$ or $\Delta (h;f,q)$ on the set $H_{\mathcal{D}}\times \mathcal{D}$.

The saddle point inequalities

$$\Delta (h;f^0)\ge \Delta (h^0;f^0)\ge \Delta (h^0;f)\forall f\in {\mathcal{D}}_f,\forall h\in H_{\mathcal{D}},$$

hold true if $h^0=h_{\mu }\left(f^0\right)$ and $h_{\mu }\in H_{\mathcal{D}}$, where $f^0$ is a solution of the constraint optimisation problem

$$\tilde{\Delta }\left(f\right)=-\Delta (h_{\mu }\left(f^0\right);f)\to inf,f\in {\mathcal{D}}_f,$$

(34)

where the functional $\Delta (h_{\mu }\left(f^0\right);f)$ is calculated by the formula

$$\Delta (h_{\mu }\left(f^0\right);f)={\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{\left|{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}{f}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{C}_{\mu ,N}^0{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }{\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda ,$$

and $C_{\mu ,N}^0\left(e^{i\lambda }\right)$ is derived from the system of equations (18) for $f^0\left(\lambda \right)$.

The saddle point inequalities

$$\Delta (h;f^0,g^0)\ge \Delta (h^0;f^0,g^0)\ge \Delta (h^0;f,g)\forall f\in {\mathcal{D}}_f,\forall g\in {\mathcal{D}}_g,\forall h\in H_{\mathcal{D}}$$

hold true if $h^0=h_{\mu }(f^0,g^0)$ and $h_{\mu }\in H_{\mathcal{D}}$, where $(f^0,g^0)$ is a solution of the constraint optimisation problem

$$\tilde{\Delta }(f,g)=-\Delta (h_{\mu }(f^0,g^0);f,g)\to inf,(f,g)\in {\mathcal{D}}_f\times {\mathcal{D}}_g,$$

(35)

where the functional $\Delta (h_{\mu }(f^0,g^0);f,g)$ is calculated by the formula

$$\begin{array}{cc}\hfill \Delta (h_{\mu }(f^0,g^0);f,g)& ={\int }_{-\pi }^{\pi }{\left|B_N^{\mu }\left(e^{i\lambda }\right)-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{\alpha }{\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda \hfill \\ & +{\int }_{-\pi }^{\pi }{\left|B_N^{\mu }\left(e^{i\lambda }\right)-A_N\left(e^{i\lambda }\right){\displaystyle \frac{1}{{(1-e^{-i\lambda \mu })}^n}}-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{\alpha }{|1-e^{-i\lambda \mu }|}^{\alpha n}g\left(\lambda \right)d\lambda ,\hfill \end{array}$$

and $H_{\mu }^0\left(e^{i\lambda }\right)$ is derived from the system of equations (23) for $(f^0\left(\lambda \right),g^0\left(\lambda \right))$.

The saddle point inequalities

$$\Delta (h;f^0,q^0)\ge \Delta (h^0;f^0,q^0)\ge \Delta (h^0;f,q)\forall f\in {\mathcal{D}}_f,\forall q\in {\mathcal{D}}_q,\forall h\in H_{\mathcal{D}}$$

hold true if $h^0=h_{\mu }(f^0,q^0)$ and $h_{\mu }\in H_{\mathcal{D}}$, where $(f^0,q^0)$ is a solution of the constraint optimisation problem

$$\tilde{\Delta }(f,q)=-\Delta (h_{\mu }(f^0,q^0);f,q)\to inf,(f,q)\in {\mathcal{D}}_f\times {\mathcal{D}}_q,$$

(36)

where the functional $\Delta (h_{\mu }(f^0,q^0);f,q)$ is calculated by the formula

$$\begin{array}{cc}\hfill & \Delta (h_{\mu }(f^0,q^0);f,q)\hfill \\ & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{\left|{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}{f}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{C}_{\mu ,N}^{0,1}{\left(e^{i\lambda }\right)}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }{\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda \hfill \\ & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{\left|{\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}{q}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}{(C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)-C_{\mu ,N}^{0,2}\left(e^{i\lambda }\right))}^{<{\displaystyle \frac{1}{\alpha -1}}>}\right|}^{\alpha }{\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}q\left(\lambda \right)d\lambda ,\hfill \end{array}$$

and $C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)$, $C_{\mu ,N}^{0,2}\left(e^{i\lambda }\right)$ are derived from the system of equations (29)-(30) for $(f^0\left(\lambda \right),q^0\left(\lambda \right))$.

The constrained optimisation problems (34), (35), (36) are equivalent to the unconstrained optimisation problems

$$\begin{array}{cc}\hfill {\Delta }_{\mathcal{D}}\left(f\right)& =\tilde{\Delta }\left(f\right)+\delta \left(f\right|{\mathcal{D}}_f)\to inf,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\Delta }_{\mathcal{D}}(f,g)& =\tilde{\Delta }(f,g)+\delta (f,g|{\mathcal{D}}_f\times {\mathcal{D}}_g)\to inf,\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\Delta }_{\mathcal{D}}(f,q)& =\tilde{\Delta }(f,q)+\delta (f,q|{\mathcal{D}}_f\times {\mathcal{D}}_q)\to inf,\hfill \end{array}$$

(39)

where $\delta \left(f\right|{\mathcal{D}}_f)$, $\delta (f,g|{\mathcal{D}}_f\times {\mathcal{D}}_g)$, $\delta (f,q|{\mathcal{D}}_f\times {\mathcal{D}}_q)$ are the indicator functions of the sets ${\mathcal{D}}_f$, ${\mathcal{D}}_f\times {\mathcal{D}}_g$, ${\mathcal{D}}_f\times {\mathcal{D}}_q$. Solution $f^0$, $(f^0,g^0)$, $(f^0,q^0)$ to this unconstrained optimisation problems are characterized by the condition $0\in \partial {\Delta }_{\mathcal{D}}\left(f^0\right)$, $0\in \partial {\Delta }_{\mathcal{D}}(f^0,g^0)$, $0\in \partial {\Delta }_{\mathcal{D}}(f^0,q^0)$, where $\partial {\Delta }_{\mathcal{D}}\left(f^0\right)$, $\partial {\Delta }_{\mathcal{D}}(f^0,g^0)$, $\partial {\Delta }_{\mathcal{D}}(f^0,q^0)$ are the subdifferentials of the functional ${\Delta }_{\mathcal{D}}\left(f\right)$ at point $f^0\in {\mathcal{D}}_f$, the functional ${\Delta }_{\mathcal{D}}(f,g)$ at point $(f^0,g^0)\in {\mathcal{D}}_f\times {\mathcal{D}}_g$, the functional ${\Delta }_{\mathcal{D}}(f,q)$ at point $(f^0,q^0)\in {\mathcal{D}}_f\times {\mathcal{D}}_q$ [23,24]. This conditions make it possible to find the least favourable spectral densities in some special classes of spectral densities ${\mathcal{D}}_f$, ${\mathcal{D}}_f\times {\mathcal{D}}_g$, ${\mathcal{D}}_f\times {\mathcal{D}}_q$.

4. Examples of Minimax Estimation for Specific Classes of Admissible Spectral Densities

4.1. Least Favorable Spectral Density in the Class ${\mathcal{D}}_f$

Consider the estimation problem without noise for the functional $A_N\xi ={\sum }_{k=0}^{N}a\left(k\right)\xi \left(k\right)$ which depends on unobserved values of the sequence $\xi \left(m\right)$ with $HS\alpha S$ increments based on its observations at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$, where the set of admissible spectral densities is defined as follows:

$${\mathcal{D}}_f=\left\{f\left(\lambda \right)|{\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)d\lambda =P\right\}.$$

Suppose that the linear functional $\Delta (h_{\mu }\left(f^0\right);f)$ with respect to the variable f is bounded in the space $L_1$. Applying the Lagrange method of indefinite multipliers to optimization problem (37), we obtain that the spectral density $f^0\in {\mathcal{D}}_f$ satisfy the equation

$${\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}{f}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{\alpha }{\alpha -1}}}{\left|C_{\mu ,N}^0\left(e^{i\lambda }\right)\right|}^{{\displaystyle \frac{\alpha }{\alpha -1}}}=\gamma .$$

From this equation we find that the least favourable spectral density is of the form

$$f^0\left(\lambda \right)={\gamma }^{{\displaystyle \frac{1-\alpha }{\alpha }}}{\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}}}\left|C_{\mu ,N}^0\left(e^{i\lambda }\right)\right|.$$

(40)

Thus, we have the following statement.

Theorem 4.  

Let the spectral density $f^0\left(\lambda \right)\in {\mathcal{D}}_f$ be determined by equation (40), where $C_{\mu ,N}^0\left(e^{i\lambda }\right)$ is derived from system of equations (18). Then it is the least favourable spectral density in the class ${\mathcal{D}}_f$ for the optimal linear estimation of the functional $A_N\xi$ based on observations $\xi \left(m\right)$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$ if it satisfies minimality condition (11), restriction on the densities in the class ${\mathcal{D}}_f$ and determines a solution of optimization problem (34). The minimax spectral characteristic $h_{\mu }\left(f^0\right)$ is determined by formula (17).

4.2. Least Favorable Spectral Densities in the Class ${\mathcal{D}}_f^{\beta }\times {\mathcal{D}}_g^{\beta }$

Consider the estimation problem with noise for the functional $A_N\xi ={\sum }_{k=0}^{N}a\left(k\right)\xi \left(k\right)$ which depends on unobserved values of the sequence $\xi \left(m\right)$ with $HS\alpha S$ increments based on observations $\xi \left(m\right)+\eta \left(m\right)$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$, where $\eta \left(m\right)$ is a $HS\alpha S$ sequence independent from $\xi \left(m\right)$. Let the set $\mathcal{D}$ of admissible spectral densities is defined as follows:

$$\begin{array}{cc}\hfill {\mathcal{D}}_f^{\beta }& =\left\{f\left(\lambda \right)|{\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)\right)}^{{\beta }_1}d\lambda =P_1\right\},\hfill \\ \hfill {\mathcal{D}}_g^{\beta }& =\left\{g\left(\lambda \right)|{\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{\left(g\left(\lambda \right)\right)}^{{\beta }_2}d\lambda =P_2\right\},\hfill \end{array}$$

where ${\beta }_1\ne 1$ and ${\beta }_2\ne 1$.

Suppose that the linear functional $\Delta (h_{\mu }(f^0,g^0);f,g)$ with respect to the variable $(f,g)$ is bounded in the space $L_1$. Applying the Lagrange method of indefinite multipliers to optimization problem (), we obtain that the spectral densities $f^0\in {\mathcal{D}}_f^{\beta }$, $g^0\in {\mathcal{D}}_g^{\beta }$ satisfy the equations

$${\left|B_N^{\mu }\left(e^{i\lambda }\right)-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{\alpha }={\gamma }_1{\left({\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}{f}^0\left(\lambda \right)\right)}^{{\beta }_1-1}$$

and

$${\left|B_N^{\mu }\left(e^{i\lambda }\right)-A_N\left(e^{i\lambda }\right){\displaystyle \frac{1}{{(1-e^{-i\lambda \mu })}^n}}-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{\alpha }{|1-e^{-i\lambda \mu }|}^{\alpha n}={\gamma }_2{\left(g^0\left(\lambda \right)\right)}^{{\beta }_2-1}.$$

From these equations we find that the least favourable spectral densities are of the form

$$\begin{array}{cc}\hfill f^0\left(\lambda \right)& ={\gamma }_1^{{\displaystyle \frac{-1}{{\beta }_1-1}}}{\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}}}{\left|B_N^{\mu }\left(e^{i\lambda }\right)-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{{\displaystyle \frac{\alpha }{{\beta }_1-1}}},\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill g^0\left(\lambda \right)& ={\gamma }_2^{{\displaystyle \frac{-1}{{\beta }_2-1}}}{|1-e^{-i\lambda \mu }|}^{{\displaystyle \frac{\alpha n}{{\beta }_2-1}}}{\left|B_N^{\mu }\left(e^{i\lambda }\right)-A_N\left(e^{i\lambda }\right){\displaystyle \frac{1}{{(1-e^{-i\lambda \mu })}^n}}-H_{\mu }^0\left(e^{i\lambda }\right)\right|}^{{\displaystyle \frac{\alpha }{{\beta }_2-1}}}.\hfill \end{array}$$

(42)

Thus, we have the following statement.

Theorem 5.  

Let the spectral densities $f^0\left(\lambda \right)\in {\mathcal{D}}_f^{\beta }$, $g^0\left(\lambda \right)\in {\mathcal{D}}_g^{\beta }$ be determined by equations (41), (), where $H_{\mu }^0\left(e^{i\lambda }\right)$ is derived from system of equations (23). Then they are the least favourable spectral densities in the class ${\mathcal{D}}_f^{\beta }\times {\mathcal{D}}_g^{\beta }$ for the optimal linear estimation of the functional $A_N\xi$ based on observations $\xi \left(m\right)+\eta \left(m\right)$ at points $m\in \mathbb{Z}\setminus \{0,1,2,\dots ,N\}$ if they satisfy minimality condition (12), restrictions on the densities in the classes ${\mathcal{D}}_f^{\beta }$, ${\mathcal{D}}_g^{\beta }$ and determine a solution of optimization problem (35). The minimax spectral characteristic $h_{\mu }(f^0,g^0)$ is determined by formula (22).

4.3. Least Favorable Spectral Densities in the Class ${\mathcal{D}}_f^{\beta }\times {\mathcal{D}}_q^{\beta }$

Consider the estimation problem with semi-noise for the functional $A_N\xi ={\sum }_{k=0}^{N}a\left(k\right)\xi \left(k\right)$ which depends on unobserved values of the sequence $\xi \left(m\right)$ with $HS\alpha S$ increments based on observations $\xi \left(m\right)$ at points $m\le -1$ and $\xi \left(m\right)+\theta \left(m\right)$ at points $m\ge N+1$, where $\theta \left(m\right)$ is a sequence with $HS\alpha S$ increments independent from $\xi \left(m\right)$. Let the set $\mathcal{D}$ of admissible spectral densities is defined as follows:

$$\begin{array}{cc}\hfill {\mathcal{D}}_f^{\beta }& =\left\{f\left(\lambda \right)|{\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}f\left(\lambda \right)\right)}^{{\beta }_1}d\lambda =P_1\right\},\hfill \\ \hfill {\mathcal{D}}_q^{\beta }& =\left\{q\left(\lambda \right)|{\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{\left({\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}q\left(\lambda \right)\right)}^{{\beta }_2}d\lambda =P_2\right\},\hfill \end{array}$$

where ${\beta }_1\ne -1/(\alpha -1)$ and ${\beta }_2\ne -1/(\alpha -1)$.

Suppose that the linear functional $\Delta (h_{\mu }(f^0,q^0);f,q)$ with respect to the variable $(f,q)$ is bounded in the space $L_1$. Applying the Lagrange method of indefinite multipliers to optimization problem (), we obtain that the spectral densities $f^0\in {\mathcal{D}}_f^{\beta }$, $q^0\in {\mathcal{D}}_q^{\beta }$ satisfy the equations

$${\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}{f}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{\alpha }{\alpha -1}}}{\left|C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)\right|}^{{\displaystyle \frac{\alpha }{\alpha -1}}}={\gamma }_1{\left({\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}{f}^0\left(\lambda \right)\right)}^{\beta -1}$$

and

$${\left({\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}{q}^0\left(\lambda \right)}}\right)}^{{\displaystyle \frac{\alpha }{\alpha -1}}}{|C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)-C_{\mu ,N}^{0,2}\left(e^{i\lambda }\right)|}^{{\displaystyle \frac{\alpha }{\alpha -1}}}={\gamma }_2{\left({\displaystyle \frac{|1-e^{-i\lambda \mu }{|}^{\alpha n}}{{\left|\lambda \right|}^{\alpha n}}}{q}^0\left(\lambda \right)\right)}^{\beta -1}.$$

From these equations we find that the least favourable spectral densities are of the form

$$\begin{array}{cc}\hfill f^0\left(\lambda \right)& ={\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}}}{\gamma }_1^{{\displaystyle \frac{\alpha -1}{-\alpha -(\alpha -1)({\beta }_1-1)}}}{\left|C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)\right|}^{{\displaystyle \frac{-\alpha }{-\alpha -(\alpha -1)({\beta }_1-1)}}},\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill q^0\left(\lambda \right)& ={\displaystyle \frac{{\left|\lambda \right|}^{\alpha n}}{|1-e^{-i\lambda \mu }{|}^{\alpha n}}}{\gamma }_2^{{\displaystyle \frac{\alpha -1}{-\alpha -(\alpha -1)({\beta }_2-1)}}}{|C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)-C_{\mu ,N}^{0,2}\left(e^{i\lambda }\right)|}^{{\displaystyle \frac{-\alpha }{-\alpha -(\alpha -1)({\beta }_2-1)}}}.\hfill \end{array}$$

(44)

Thus, we have the following statement.

Theorem 6.  

Let the spectral densities $f^0\left(\lambda \right)\in {\mathcal{D}}_f^{\beta }$, $q^0\left(\lambda \right)\in {\mathcal{D}}_q^{\beta }$ be determined by equations (43), (44), where $C_{\mu ,N}^{0,1}\left(e^{i\lambda }\right)$ and $C_{\mu ,N}^{0,2}\left(e^{i\lambda }\right)$ are derived from system of equations (29)-(30). Then they are the least favourable spectral densities in the class ${\mathcal{D}}_f^{\beta }\times {\mathcal{D}}_q^{\beta }$ for the optimal linear estimation of the functional $A_N\xi$ based on observations $\xi \left(m\right)$ at points $m\le -1$ and $\xi \left(m\right)+\theta \left(m\right)$ at points $m\ge N+1$ if they satisfy minimality conditions (11), restrictions on the densities in the classes ${\mathcal{D}}_f^{\beta }$, ${\mathcal{D}}_q^{\beta }$ and determine a solution of optimization problem (36). The minimax spectral characteristic $h_{\mu }(f^0,q^0)$ is determined by formulas (27), (28).

5. Discussion

In this article, results of investigation of the classical and minimax estimation problems for a class of stochastic sequences with harmonizable symmetric $\alpha$-stable nth increments are presented. Extrapolation, interpolation and filtering problems were earlier investigated for the linear functionals from the harmonizable symmetric $\alpha$-stable stochastic sequence and from the stochastic sequence with stationary increments. Thus, the current work extends the random processes estimation theory to a new class of non-stationary sequences. The further directions of the studies are extrapolation and filtering problems for the considered sequences as well as estimation of functionals of continuous time random processes with harmonizable symmetric $\alpha$-stable nth increments.