Partial sum process to check regression models with multiple correlated response : with an application for testing a change point in profile data

We investigate a data set describing the quality of a production process. By the information of these data it has to be decided whether the quality is constant or whether the quality changes. Our null hypothesis is that the quality is constant which is a linear regression. In practice it is popular to investigate the partial sums of the least squares residuals to look for changes in linear regression. The partial sums of the least squares residuals can be embedded into the class of continuous functions. By this procedure we obtain a stochastic process with continuous paths. It is called residual partial sum process. If the number of observations is large enough a projection of the Brownian motion can be considered as approximation (with respect to weak convergence) of the residual partial sum process. This projection of the Brownian motion can be used to establish nonparametric tests of Cramér-von Mises and Kolmogorov-Smirnov type to test for changes in linear regression. We use this procedure to test the data for constant quality