Limit theorems for quadratic forms with applications to Whittle's estimate
type: Item
instance of: scholarly article
title: Limit theorems for quadratic forms with applications to Whittle's estimate
zbMATH Open document ID: 0940.60037
author: Lajos Horváth
author: Q. M. Shao
published in: The Annals of Applied Probability
publication date: 2000-07-13T00:00:00Z
review text: This paper corrects the limit distribution of a quadratic form statistic previously announced by \textit{J. Beran} and \textit{N. Terrin} [Biometrika 83, No. 3, 627-638 (1996; Zbl 0866.62055)]. This difference of quadratic forms can be used in determining the presence of a change-point in the long-memory parameter of a dependent sequence of random variables. The underlying data, \(X_i\), are assumed to be a moving average of iid variables. Define \[ Q^*(x,y) =\sum_{[x]<i,j\leq [y]}b(i-j)\{X_iX_j-r(i-j)\} \] for a suitable sequence of constants \(b(k)\) and where \(r(k)\) denotes the covariance of \(X_0\) and \(X_k\). Define \(Q(x)=Q^*(x,\infty)\) and \[ Z_n(t)=n^{-1/2}t(1-t)\{Q(nt)/nt - Q^*(nt,n)/n(1-t)\}. \] Under decay conditions on the moving average weights and the constants \(b\), the paper establishes a strong approximation of \(Q(x)\) by a Brownian motion, and of \(Z_n\) by a sequence of independent Brownian bridges. It then obtains a strong law of large numbers for \(Q(n)\). The paper examines the case where the moving average coefficients for \(X_k\) depend on a finite-dimensional parameter, \(\lambda_k\). These are assumed to be constant up to a certain point, and then equal to a different constant after that change-point. Splitting the data at \(k\), the Whittle estimate of \(\lambda\) before \(k\) is \(\widehat{\lambda}_k\): that after, \(\widetilde{\lambda}_k\). Theorem 2.1 obtains that under the null hypothesis that there is no change-point, the normalized difference \(\widehat{\lambda}_{nt}-\widetilde{\lambda}_{nt}\) converges weakly to a scaled Brownian motion. Theorem 2.2 provides a similar result for weighted metrics. Theorem 2.3 provides upper tail critical points. The proofs decompose the quadratic forms into martingales and smaller terms.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 1319822
zbMATH Keywords: quadratic forms
zbMATH Keywords: change-point
zbMATH Keywords: long-memory parameter
reviewed by: Andre R. Dabrowski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1029962600
OpenAlex ID: W2092926041
cites work: On the Asymptotic Distribution of the Autocorrelations of a Sample from a Linear Stochastic Process
cites work: On bilinear forms in Gaussian random variables and Toeplitz matrices
cites work: Testing for a change of the long-memory parameter
cites work: Q5560061
cites work: Q5288953
cites work: Q4348180
cites work: Q3326516
cites work: Order-of-magnitude bounds for expectations involving quadratic forms
cites work: Central limit theorems for quadratic forms in random variables having long-range dependence
cites work: A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate
cites work: Q3911791
cites work: The asymptotic theory of linear time-series models
cites work: On Limit Theorems for Quadratic Functions of Discrete Time Series
cites work: Strong approximation for cross-covariances of linear variables with long-range dependence
cites work: Q4084474
DOI: 10.1214/AOAP/1029962600
description: scientific article; zbMATH DE number 1319822