On the mean estimation of variance for an asymptotically stationary stochastic process with bounded duration
type: Item
instance of: scholarly article
title: On the mean estimation of variance for an asymptotically stationary stochastic process with bounded duration
zbMATH Open document ID: 0935.93056
author: V. E. Kulikov
published in: Journal of Computer and Systems Sciences International
publication date: 2000-05-04T00:00:00Z
review text: A stable linear stochastic dynamic system \[ \dot x(t)= Ax(t)+ B\xi(t),\quad t_0\leq t\leq t_0+ T,\quad x(t_0)= x_0\tag{1} \] is considered. Here \(x(t)\in \mathbb{R}^n\), \(A\) is a nonsingular matrix, \(\xi(t)\) is \(r\)-dimensional white noise, \(B\) is an \(n\times r\) matrix of distribution of control, and \(x_0\) is a random vector of initial states with \(Ex_0= 0\). By a single realization of the random process on the interval \([t_0,t_0+ T]\), the empirical estimation of the covariance matrix \(E[x(t)x^T(t)]\) is constructed, \[ \widehat D(t_0, T)= T^{-1}\cdot \int^{t_0+ T}_{t_0} [x(t)-\widehat m(t_0, T)]\cdot [x(t)-\widehat m(t_0, T)]^T dt, \] with \[ \widehat m(t_0, T)= T^{-1}\cdot \int^{t_0+ T}_{t_0} x(\lambda) d\lambda. \] An expression for the mean value of \(\widehat D(t_0, T)\) is derived for the process (1) in its nonstationary state. For the stationary state, this was done by \textit{A. N. Tikhonov} and \textit{V. N. Kharisov} [Statistical analysis and synthesis of radiotechnical devices and systems (Russian), Moscow: Radio i Svyaz' (1991)]. Two examples are given: a) the problem of the sliding average in time for a one-dimensional casual asymptotically stationary process, and b) estimation of the variation of the relief heights in foothill regions.
Mathematics Subject Classification ID: 93E03
zbMATH DE Number: 1274252
zbMATH Keywords: nonstationary state
zbMATH Keywords: mean estimation of variance
zbMATH Keywords: stable linear stochastic dynamic system
reviewed by: A. Kukush
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1274252