Multivariate Recursive Residuals Partial Sums Processes and Its Application in Model Check for Multiresponse Regression

摘要

In this paper asymptotic test in multivariate regression based on set-indexed partial sums of the vector of recursive residuals is proposed. The limit process is derived for multivariate nonparametric regression with localized vector of regression functions under an equally spaced experimental design on a closed rectangle. Under mild condition it is shown that independent to the assumed model, the partial sums processes converges to a vector of trends plus the multivariate set-indexed Brownian sheet. The trend vanishes simultaneously when the hypothesis is true living the multivariate set-indexed Brownian sheet as the only limit process. The finite sample size behavior of the power functions of Kolmogorov-Smirnov (KS) and Cramer-von Mises (CM) type tests are investigated by simulation. It is shown that for testing multivariate polynomial model of low order the CM test seems to have larger power than the KS test has. The application of the test method in the empirical model building of corn plants data and its comparison with the classical test using Wilk's lambda statistic is also demonstrated.