A Change-Point Problem and Composite Likelihood Inference in Time Series Models

This thesis consists of essentially two parts, both of which are related to likelihood inference in time series models. The first part addresses a hypothesis testing problem about distinguishing between a short-memory time series with structural change and a long-memory time series without structural change. The second part is a study of composite likelihood inference in time series. While these two parts seem quite different, they are connected in a chapter that considers the application of composite likelihood on the problems in change point analysis.;The first part of the thesis develops a likelihood ratio based test for discriminating between a short-memory time series with structural change and a long-memory time series without structural change. Under the null hypothesis, the time series consists of two segments of short-memory time series with different means, and possibly different covariance functions. The location of the structural change is unknown. Under the alternative hypothesis the time series has no structural change but rather is long-memory. The likelihood ratio statistic is defined as the normalized log-ratio of the Whittle likelihood between the change-point model and the long-memory model, which is asymptotically normally distributed under the null. Berkes et al.(2006) proposed a test based on the CUSUM statistics for discriminating between change-point and long-memory models. The CUSUM test assumes that under the null hypothesis the time series has a shift in mean at an unknown location. Other properties of the time series such as correlation function and marginal distribution are assumed unchanged. The likelihood ratio test provides a parametric alternative to the CUSUM test proposed by Berkes et al. (2006). Moreover, the likelihood ratio test is more general than the CUSUM test in the sense that it is applicable to changes in other marginal or dependence features other than a change-in-mean. Furthermore, the likelihood ratio test tends to have higher power than the CUSUM test on detecting the long-memory Fractionally Integrated Autoregressive Moving Average (ARFIMA) model. We show its good performance in simulations and apply it to two data examples.;The second component in this thesis discusses composite likelihood inference in time series. In modern statistical analysis, one often encounters complicated models for complex data. Exact likelihood may not be a real option in these cases because of computational difficulties or the unavailability of theoretical asymptotic properties. Recently there has been considerable development in composite likelihood theory, which tackles the above problems by working on some approximations likelihood theory in some time series likelihood estimation procedures for linear time series models are described. The to the exact likelihood. This thesis attempts to study the application of composite models. The asymptotic properties of pairwise estimators are shown to be strongly consistent and asymptotically normal. This covers Autoregressive Moving Average (ARMA) as well as fractionally integrated ARMA (ARFIMA) models with the fractional integration parameter d < 0.5. A comparison between using all pairs and consecutive pairs of observations in defining the composite likelihood is given. In particular, when all possible pairs of observations are used in defining the likelihood, the estimation for ARMA model is still consistent. However, the estimation for ARFIMA model is only consistent for d < 0.25 but not consistent for d ≥ 0.25. Application of pairwise likelihood to a popular nonlinear model for time series of counts is also considered.;Finally, this thesis suggests some possible applications of pairwise likelihood to change-point analysis, including a connection between pairwise likelihood and distinguishing between short-memory change-point model and long-memory model. Since pairwise likelihood can be applied to complicated models where exact likelihood method is not feasible, e.g. time series with latent process, the use of pairwise likelihood can broaden the scope of applicability to problems in change-point analysis. For instance, estimation of multiple change-points in Poisson autoregressive models or testing for change-points in stochastic volatility models may be possible using a composite likelihood approach.