Residual partial sum limit process for regression models with applications to detecting parameter changes at unknown times

Limit processes for sequences of stochastic processes defined by partial sums of linear functions of regression residuals are derived. They are Gaussian and are functions of standard Brownian motion. Cramér-von Mises type functionals defined on the partial sum processes are shown to converge in distribution to the same functionals defined on the limit processes. This result is then applied to derive the asymptotic forms of two-sided change detection statistics for linear regression models. These are derived for a variety of weight sequences and are shown to involve sums of Cramér-von Mises type stochastic integrals. Finally a methodology is developed to derive distributions of these stochastic integrals for the case of harmonic regression. This methodology is applicable to more general situations.