The Study Of Consistency For CUSUM-type Statistics Of Change-point Models And Its Applications

The research on the change point problem comes from industrial quality control and management.For example,to detect the change of an important index in the production process,a detection index threshold is set for this detection index.If the detected index exceeds the set threshold,the detection system will give an early warning in time,so as to ensure product quality and reduce the loss caused by unqualified product quality.At this time,the time when the indicator exceeds the set threshold in the detection process is called the change point time or the change point location.The change point detection has been applied in many fields such as economy,finance,meteorology and management,etc.The change point detection models include the mean change and variance change,and the cumulative sum(CUSUM)method is one of the commonly used to study the change point detection.At present,there are many results of mean change-point models based on normal data or independent data.However,in practice,it is difficult for the data to meet the normality or independence requirements.Therefore,this thesis will study the mean change-point models under the dependent data such as positive associated(PA)sequence.We study the consistency of the CUSUM-type change-point statistics for the mean change-point models.Under some weak conditions,some theoretical results such as the limit distribution and convergence rate are obtained for the change-point statistics,which extend some existing research results.In order to test our theoretical results,some simulation as empirical sizes,empirical powers and convergence are presented in this thesis.As important applications,we use CUSUM statistics to do the mean change-point analysis for the monthly temperature changes in Quebec City and the return changes for Jingdong stock and Tesla stock.Below we give the main research content of this thesis.In Chapter 1,we introduce the background knowledge,research methods and re-search status of change-point test.In addition,we give some important definitions and lemmas,including the definition of PA sequences,the properties of PA sequences and the Donsker invariance principle.In Chapter 2,we investigate the limit distribution study for the CUSUM-type change-point test statistic.In order to test whether there is a change-point or not,we give a CUSUM-type statistic for the mean change-point models.Under some regular conditions and the mean change?_nis small enough to satisfy n^1/2?_n=o(1),we use the Donsker invariance principle of the PA sequence to obtain the limit distribution for our CUSUM-type statistic.Since the CUSUM-type statistic contains unknown parameters of the variance term,we use the sample variance and sample covariance to estimate it.With help of consistent estimator of the variance term,we also obtain the limit distribution for the CUSUM-type statistic.Using the results of the limit distribution theory,we can judge whether the mean change of the mean change-point model satisfies n^1/2?_n=o(1)or not.At the significance level,if it is judged that there exists some big mean change,then we give a CUSUM-type estimator for the mean change point location.In order to test our CUSUM-type change-point test statistics,some simulations of empirical size and power were presented under normal data and skewed data.To compare empirical size and power,the existing means change point test methods are also carried out.The simulations show that our empirical size and power of our method have the expected result.In chapter 3,we study the convergence rate of the CUSUM-type change-point test statistic.In chapter 2,we can use the CUSUM-type statistic of limit distribution to judge whether there exists a big mean change or not.So in this chapter,we study the consistency of the CUSUM-type change-point location ratio estimator when there exists a big mean change amount in mean change-point models.By using the H?ajek-R?enyi-type inequality,we obtain some convergence rates for the change-point location ratio estimator,which extend the results of Kokoszka and Leipus(Stat.Probab.Lett.,1998,40,385-393).In addition,we carry out the convergence simulations for the CUSUM-type change point location ratio estimator under normal data and skew data.The simulation results are consistent with our theoretical convergence rates.In Chapter 4,we carried out real data analysis of mean change-point models.For example,we give the mean change-point model for the monthly average temperature of June and October for Quebec City,Canada.Using our CUSUM-type change point location estimator and other existing methods,we successfully detect the same change-point location.Meanwhile,we give the mean change-point models for the changes of return of JD stock and Tesla stock.With the help of our CUSUM-type change point estimators and other existing estimators,we find the same change-point locations.In addition,some economic reasons are illustrated for these change point locations.In Chapter 5,we summarize our works in this thesis and give some discussion for the future research of mean change point models.For example,we can consider the study of limit distribution and convergence rates of the CUSUM-type statistic of the mean change-point models under other dependent sequences.Furthermore,Shiryaev(Stochastic Dis-order Problems,Springer,2019)studied the Black-Scholes model with mean shift.So we will pay attention to the study of change-point models in mathematical finance and econometrics.