정수값을 가지는 시계열 모형에서의 모수 변화 검정과 로버스트 추정방법

Abstract

학위논문 (박사)-- 서울대학교 대학원 : 통계학과, 2014. 2. 이상열.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.Abstract i List of Tables viii List of Figures ix 1 Introduction 1 2 Reviews 6 2.1 The integer-valued time series models . . . . . . . . . . . . . . . . . . 6 2.2 The cumulative sum (CUSUM) test . . . . . . . . . . . . . . . . . . . 8 2.3 The minimum density power divergence estimator (MDPDE) . . . . . 10 3 Parameter Change Test for Random Coefficient Integer-Valued Autoregressive Processes with Application to Polio Data Analysis 13 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Estimation for RCINAR(1) models . . . . . . . . . . . . . . . . . . . 15 iii 3.3 Change point test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5 Real data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Parameter Change Test for Poisson Autoregressive Models 41 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Estimation for Poisson autoregressive model . . . . . . . . . . . . . . 42 4.2.1 Poisson autoregressive model . . . . . . . . . . . . . . . . . . . 42 4.2.2 INGARCH(1,1) model . . . . . . . . . . . . . . . . . . . . . . 46 4.3 Change point test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3.1 Estimates-based CUSUM test . . . . . . . . . . . . . . . . . . 48 4.3.2 Residual-based CUSUM test . . . . . . . . . . . . . . . . . . . 52 4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4.1 INGARCH(1,1) model . . . . . . . . . . . . . . . . . . . . . . 53 4.4.2 Poisson threshold autoregressive model . . . . . . . . . . . . . 57 4.5 Real data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.8 Supplementary material . . . . . . . . . . . . . . . . . . . . . . . . . 80 5 Minimum Density Power Divergence Estimator for Poisson Autoreiv gressive Models 87 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2 MDPDE in Poisson autoregressive models . . . . . . . . . . . . . . . 88 5.3 Asymptotic properties of MDPDE . . . . . . . . . . . . . . . . . . . . 90 5.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.5 Real data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 BibliographyDocto