Statistical Inference In The Jump Change-point Estimator

Change-point problem has been a hot topic in statistics since the1970s. It is widely used in industrial quality control, economic, financial, medical, computer and other fields. The so-called change-point refers to the observation values or data follow two different models before or after a position or moment Ï„0in a sequence or process. In this thesis, we investigate the Op convergence rate and asymptotic distribution of the jump change-point estimator that given by sliding window method, and are divided in four chapters to introduce.In chapter1, we first give a brief overview of the origin of change-point problem and common research content and methods. Then we introduce the research situation of different change-point problem using different methods. Furthermore, we describe the research of the application of change-point in different field. Finally we describe the research situation of the jump change-point and the convergence rate and asymptotic distribution of change-point estimator.The second chapter investigates the change-point model Xi=a+θI{[nÏ„0]<i<n)+εi, i=1,2,…,n which have only one jump change-point Ï„0,under the assumption that εx, ε2,…,εn is an independent and identically distributed random variable sequence with0mean value and finite variance. We investigate the Op convergence rate of change-point estimator Ï„ under local alternative hypothesis based on the change-point estimator given by sliding window method, this is, the jump θ depends on the size of sample n, hence, we denote it as θn, and θn satisfises θâ†'0as nâ†'∞, then we have when the window width to meet certain conditions.Chapter3gives a further research of asymptotic distribution of change-point estimator. Under the condition of local alternative hypothesis, we give that the asymptotic distribution of the change-point estimator Ï„ is sup(W(s)-|s|) when σ2is known, where W(s) is a two-side Brown motion which is defined in (-∞,+∞), and we prove that with the slip window method. We use σ2n and θ2(m) instead when σ2is unknown, which are the estimators of σ2and θn. The summary for the full thesis is given and the future research tendcy ofchange-point is discussed in chapter4.