Change Point Problems in Regression

摘要

This dissertation focuses on the problem of testing for a change in the regression model when errors are independently, normally distributed with constant, known or unknown variance. In both models, the author considers the likelihood ratio test (LRT) as the problem of the boundary crossing by the discrete stochastic process and studies problems such as approximations to significance levels, powers, and confidence regions for a change point. First of all, the proposes a modified LRT and discusses asymptotic properties of test statistics in cases of random and fixed independent variables. In both cases, the author derives analytical approximations to significance levels. When the independent variables are random, the limiting distribution of the modified LRS is a function of a Brownian motion and approximations in Siegmund are used. For fixed independent variables, the limiting distribution involves a Gaussian process with nondifferentiable sample paths. In this case, an approximation is derived assuming the known variance and mild conditions about the empirical distribution of the independent variable, using the argument in Leadbetter, Lindgren and Rootzen, modified for discrete time by Hogan and Siegmund. In Model-1, we are also concerned with the power of the LRT and confidence regions for a change point. Numerical approximations of significance levels and powers of the LRT and the results of corresponding Monte Carlo experiments are obtained. (KR)