Kullback-Leibler Approach to CUSUM Quickest Detection Rule for Markovian Time Series

Abstract

Optimality properties of decision procedures are studied for the quickest detection of a change-point of parameters in autoregressive and other Markov type sequences. The limit of the normalized conditional log-likelihood ratios is shown to exist for Markov chains satisfying the ergodic theorem of information theory. Then closed-form expressions for this limit are derived by making use of the time average rate of Kullback-Leibler divergence. The good properties of the detection procedures based on a sequential analysis approach are proven to hold thanks to geometric ergodicity properties of the observation processes. In particular, the window-limited CUSUM rule is shown to be optimal for detecting the disruption point in autoregressive models. Sparre Andersen models are specifically studied.     

Data Driven Order Selection for Projection Estimator of the Spectral Density of Time Series with Long Range Dependence

Fractional exponential (FEXP) models have been introduced by Robinson (1991) and Beran (1993) to model the spectral density of a covariance stationary long-range dependent process. In this class of models, the spectral density f ( x ) of the process is decomposed as f ( x ) = |1 − exp( ix )|^{−2 d} f _*( x ), where f _*( x ) accounts for the short-memory component. In this contribution, FEXP models are used to construct semi-parametric estimates of the fractional differencing coefficient and of the spectral density, by considering an infinite Fourier series expansion of log f _*( x ). A data-driven order selection procedure, adapted from the Mallows' C _p procedure, is proposed to determine the order of truncation. The optimality of the data-driven procedure is established, under mild assumptions on the short-memory component f _*( x ). A limited Monte-Carlo experiment is presented to support our claims.     

Prediction Interval for Autoregressive Time Series via Oracally Efficient Estimation of Multi‐Step‐Ahead Innovation Distribution Function

A kernel distribution estimator (KDE) is proposed for multi‐step‐ahead prediction error distribution of autoregressive time series, based on prediction residuals. Under general assumptions, the KDE is proved to be oracally efficient as the infeasible KDE and the empirical cumulative distribution function (cdf) based on unobserved prediction errors. Quantile estimator is obtained from the oracally efficient KDE, and prediction interval for multi‐step‐ahead future observation is constructed using the estimated quantiles and shown to achieve asymptotically the nominal confidence levels. Simulation examples corroborate the asymptotic theory.