Statistical Inference For The Variance Change In Measurement Error Model

Change-point is one of the main research field in statistics. It is known that Change Point Problems began in Page(1954) who published a thesis about the con-tinuous sampling in Biometrika. After the late1960s, more statisticians devote into this field of research, and a lot of theoretic and applied papers about change Point were published. Especially the last three decades, theory and applications of the Change Point Problems have a rapid development. Krisnaiah and Miao(1988), Csorgo and Horvath(1997), Chen and Gupta(2000), Wu(2005)are the theoretical research Sum-mary of the past three decades in this area. In the application, the change point problem is no longer limited in industrial control, it is widely applied in the economic, financial, medical and computer ect.Most of literatures are focus on the study of location parameters change point, especially on the study of the mean change point, the literatures about the study of variance change point are relatively much less. In this thesis, when knowing that there is one variance change point in measurement error model, we make the estimator and study the consistency, convergence rate of the change point estimator in different meth-ods, we further study the asymptotic distribution of the estimator under the condition of the local alternative hypothesis, and make comparison with the existing results by simulation.This thesis is divided into four chapters. In the first chapter, we give a brief overview of the change point problems, and elaborate the common three research ar-eas in change point problems, describes the parameters and non-parametric methods for dealing with the change point. Especially, in the first chapter we highlights the application of U-statistic method in the statistical inference for the change point and describe the detection method for variance change point in detail.In chapter2, we first introduce the measurement error model and the measurement error model with a variance change point considered in this thesis. where{Xj} are the unable observed i.i.d variables, Wj are the contaminated versions of the independent variables Xj with measurement error ζj,ζj are independent normal variables with mean0and ζj are independent with Xj, j=1,..., n. when there is only one change point in the series of variances of{ζj}, i.e., where σ12≠σ22and ko is unknown. In this chapter, an estimate of a change point in variance of measurement errors (ME) is given in terms of characteristic functions when the variances are known. Its modification is also given for the case that the variances are unknown. In addition, the consistency and convergence rates of the estimator and its modification are investigated. The simulation study shows that the proposed estimators perform well.In chapter3,we construct a U-statistics to estimate the variance change by char-acteristic functions. Taking the advantage of the nature of the "U-statistics ", the consistency and conver-gence rates of the estimator will be investigated assuming that there is only one change point in the series of variances of{ζj}-In addition, the asymptotic distribution of the change point estimate are researched under the condition of the local alternative hypothesis that is ηn=|σ22-σ12|�'0(n�'0). we have:nρn2(τn-τ*)�'arg maxs V(s).where: λ1,λ2are depend on the second moment of the observation wi.In chapter4, we construct a CUSUM type statistics to estimate the variance change by characteristic functions.The consistency and convergence rate of change point estimate are presented. In addition, we prove that the asymptotic distribution of the change point estimate can be expressed as a bilateral Brownian motion under the condition of the local alternative hypothesis.