Detecting Structural Changes In Statistical Models Sequentially

In nature or society,there exists structural changes phenomena everywhere.The researches on the thesis are very valuable for statistical theories and applications.Analysis of structural changes provides statistical tools to study the phenomena.Statistical procedures in structural changes analysis can further be divided into two categories:historical detection and on-line detection.We will concentrate on the latter and study how to detect structural changes in three classes of statistical models:Firstly,sequential detection of structural changes is studied in structural equations models.Based on GMM,W-CUSUM(weighted cumulative sum) process is constructed. The process is proved to converge to p-dimentional Brownian bridge under null and converge into p-dimentional Brownian bridge with a "drift" under alternative.By taking a homogeneous and continuous functional on W-CUSUM process,a large class of test statistics are proposed.Provided that "drift" doesn't equal zero,the test statistics have nontrivial local power under local alternative of orderδ=1/2 and diverges under nonlocal alternative of order 0≤δ＜1/2.It is verified that the famous Cramer-Mises statistic and Lagrange-multiplier statistic are included in the class.In the case of local structural changes,W-MOSUM(weighted moving sum) process is constructed and proved to converge to h-increment(with step size h) of the Brownian bridge based on a preliminary theorem which describes convergence of the increment of a stochastic process on D[0,τ]^p. Under alternative hypothesis,W-MOSUM process converges to a Brownian bridge with a "drift".By taking a homogeneous and continuous functional on W-CUSUM process, we propose a large class of test statistics again and conclude some asymptotic properties similar to the case of W-CUSUM.Finally several Mento Carlo simulations on the two classes of statistics are performed which indicates the efficiency of our methods.Secondly,sequential detection of structural changes is studied in generalized linear models.Based on quasi-ML scores,W-CUSUM and W-MOSUM processes are constructed and proved to converge to a Brownian bridge and its h-increment correspondingly. By taking "double-max" functional on the W-CUSUM process,we constructed a test statistic and obtain its limit distribution and its exact probability formula.We also prove that the statistic is consistent under alternative and obtain the relationship between "change-point location" and "delay period".By taking "double-max" functional on the W-MOSUM process,we construct another test statistic again and gain its limit distribution and its non-explicit expression of probability formula.Then the consistency is also proved.In the subsection of simulation,we tabulate the nominal critical value for the "double-max" functional statistic and also verify the efficiency of our methods by several simulations.In the special case of GLM—normal linear model with classical link function,we compare our methods with existing ones and our methods perform more excellently.Finally,sequential detection of structural changes is studied in nonparametric regression models o Based on Nadaraya-Watson estimator of regression function,we calculate the residual series and construct "residual cumulative sum process" with weight values of kernel estimator of "long-run average density function".Under null hypothesis, the process converges to a standard Brownian bridge.By taking a homogeneous and continuous functional on the process,a large class of test statistics are constructed. We also prove their limit distributions and their corresponding construction methods of "critical values".At the same time,the process converges to a Brownian bridge with a "drift" under two alternative hypothesis,H_1A or H_1B.Provided that "drift" doesn't equal zero,they have nontrivial local power under local alternative of orderδ=1/2 and diverges under nonlocal alternative of order 0≤δ＜1/2.We investigate our distribution theory with a Monte Carlo simulation which indicates the applicability of our methods.