Comparing the marginal densities of two strictly stationary linear processes
type: Item
instance of: scholarly article
title: Comparing the marginal densities of two strictly stationary linear processes
zbMATH Open document ID: 1467.62143
author: Paul Doukhan
author: Ieva Grublytė
author: Laurence Reboul
author: D. Pommeret
published in: Annals of the Institute of Statistical Mathematics
publication date: 2021-05-25T00:00:00Z
review text: A nonparametric test is proposed in this paper to compare the marginal densities of two independent and strictly stationary linear time series. This topic is not only of theoretical interest but of relevance in several scientific areas that use time series to analyze their own topics. Let's see a brief introduction to the mathematics of the paper. Let \((\Omega,F,P)\) be a probability space, for each \(t\in\mathbb{Z}\) let \(X_{t}:\Omega\to\mathbb{R}\), \(Y_{t}:\Omega\to\mathbb{R}\) be random variables such that \(X=(X_{t})_{t\in\mathbb{Z}}\) and \(Y=(Y_{t})_{t\in\mathbb{Z}}\) are strictly stationary time series having finite second order moments. Let \((\alpha_{j})_{j\in\mathbb{Z}}\) and \((\beta_{j})_{j\in\mathbb{Z}}\) be sequences of positive real numbers such that \[ \begin{aligned} \alpha_{j}&/|j|^{\delta_{X}-1} \xrightarrow[j\to\infty]{} c_{\alpha}>0\text{ with }0<\delta_{X}<1/2, \\ \beta_{j}&/|j|^{\delta_{Y}-1} \xrightarrow[j\to\infty]{} c_{\beta}>0\text{ with }0<\delta_{Y}<1/2. \end{aligned} \] In this paper it is assumed that \[ X_{t}=\sum_{j<t}\alpha_{t-j}\varepsilon_{j},\quad Y_{t}=\sum_{j<t}\beta_{t-j}\eta_{j},\quad t\in\mathbb{Z}, \] where \((\varepsilon_{j})_{j\in\mathbb{Z}}\) and \((\eta_{j})_{j\in\mathbb{Z}}\) are sequences of identically distributed independent random variables such that \[ E(\varepsilon_{j})=0=E(\eta_{j}),\quad j\in\mathbb{Z}. \] That is \(X\) and \(Y\) are long-range-dependent (LRD) time series. It is assumed, for simplicity's sake, that \(\delta_{Y}\leq\delta_{X}\). Let \(f_{X}\) and \(f_{Y}\) be densities of \(X_0\) and \(Y_0\) respectively, with respect to the Lebesgue measure on \(\mathbb{R}\), say \(\lambda\). These functions are not assumed to be known, but it is assumed that \[ \int f_{X}^2\cdot d\rho<\infty\text{ and }\int f_{Y}^2\cdot d\rho<\infty, \] where \(\rho\) is a given probability over \(\mathbb{R}\), which has a density with respect to \(\lambda\). Note that since \(X\) and \(Y\) are strictly stationary \(f_{X}\)…
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62G10
Mathematics Subject Classification ID: 62M07
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 62P10
Mathematics Subject Classification ID: 92B25
zbMATH DE Number: 7350850
zbMATH Keywords: linear processes
zbMATH Keywords: local Whittle estimator
zbMATH Keywords: long memory
zbMATH Keywords: Schwarz's rule
zbMATH Keywords: smooth test
zbMATH Keywords: strictly stationary process
reviewed by: Oscar Bustos
describes a project that uses: longmemo
MaRDI profile type: Publication
full work available at URL: s10463-019-00730-6
OpenAlex ID: W2969730655
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Wikidata QID: Q127371155
DOI: 10.1007/S10463-019-00730-6
description: scientific article; zbMATH DE number 7350850