Limit theorems for quadratic forms with applications to Whittle's estimate
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.
scientific article; zbMATH DE number 1319822