Parameter Change Test and Robust Estimation in Integer-Valued Time Series Models

Abstract

In this thesis, we consider the parameter change test and the robust estimation for integer-valued time series models. First, we consider the problem of testing for a parameter change in a first order random coefficient integer-valued autoregressive (RCINAR(1)) model. For a test, we employ the cumulative sum (CUSUM) test based on the conditional least-squares(CLS) and modified quasi-likelihood(MQL) estimators. It is shown that under regularity conditions, the CUSUM test has the same limiting distribution as the supremum of the squares of independent Brownian bridges. The CUSUM test is then applied to the analysis of the monthly polio counts data set. Second, we consider the problem of testing for a parameter change in Poisson autoregressive models. We suggest two types of CUSUM tests: estimates-based and residual-based tests. We first demonstrate that the conditional maximum likelihood estimator (CMLE) is strongly consistent and asymptotically normal and construct the CMLE-based CUSUM test. It is shown that under regularity conditions, its limiting null distribution is a functional of independent Brownian bridges. Next, we construct the residual-based CUSUM test and derive its limiting null distribution. Simulation results are provided for illustration. A real data analysis is performed for the polio incidence data and campylobacterosis infections data. Finally, we study the robust estimation for Poisson autoregressive models. As a robust estimator, we consider a minimum density power divergence estimator (MDPDE). It is shown that under regularity conditions, the MDPDE is strongly consistent and asymptotically normal. We perform a simulation study and a real data analysis to compare the proposed estimator with MLE.