An F test for linear models with functional responses

Linear models where the response is a function, but the predictors are vectors are considered. A functional F test for choosing among two nested func- tional linear models is developed. Its null distribution is derived and a convenient approximation is presented. A simple way to test individual predictors is presented. The test is applied to some data from Ergonomics and compared to some competing tests. The ability to detect certain types of differences between models is explored. A simulation study is conducted to assess the size and power of the tests. β(t )=( X T X) −1 X T Y (t), where X is the usual n × p design matrix while Y (t) is a vector of functions of length n formed from the yi(t)'s. Eachi(t) is an independent, mean zero, Gaussian stochastic process with a covariance function r(s, t )t hat can be estimated by ˆ(s, t )= 1