A Study On Change-point Detection And Statistical Diagnosis Of AR(1) Model

Time series models have always been the focus and emphasis of statisticians,but for the complexity of the problems,usually a simple time series model is not enough to solve the real problem.In practical application,when the observed time series span a long time,the random variables forming the time series may change due to the external environment or internal factors,resulting in the change-point problem or outliers and strong influence points in the observed data,which requires the change-point detection and statistical diagnosis of the data points of the model to determine the rationality of the model.Therefore,this thesis studied the parameter estimation problems of AR(1)model through maximum likelihood(or quasi-likelihood)method,and further studied the change-point detection problems of the model and the detection problems of outliers or strong influence points based on Pena distance statistics.The research contents of this thesis are summarized as follows:Firstly,the statistical inference problems of AR(1)model with single change-point are studied.Based on the maximum likelihood(or quasi-likelihood)method,we provide the parameter estimation expression and the necessary and sufficient condition for the existence of the estimate with probability 1 for the model,and further discuss the consistency condition for the estimation of the autocorrelation coefficient.And the asymptotic distribution of autocorrelation coefficient maximum likelihood(or quasilikelihood)estimation under the conditions is obtained.Based on these,the hypothesis test on whether there is a change-point in the models and the increment of autoregressive coefficients are discussed.The results of numerical simulation and empirical analysis show that the estimation of change-point AR(1)model based on maximum likelihood(or quasi-likelihood)method has the asymptotic property of converging to true value,the hypothesis test constructed by the increment of autocorrelation coefficient can effectively detect the change-point.Secondly,the statistical diagnosis problems of AR(1)model based on Pena distance are studied.Based on maximum likelihood(or quasi-likelihood)method we discussed the maximum likelihood(or quasi-likelihood)estimation of AR(1)model parameters and the consistency of the estimates,We further obtained the expression and asymptotic distribution of the Pena distance statistic.The results of numerical simulation and empirical analysis show that the estimation of AR(1)model based on maximum likelihood(or quasi-likelihood)method has the asymptotic property of converging to true value,and the Pena distance diagnostic statistics can effectively find out the strong influence points or outliers in the model.