Performing Hypothesis Tests on the Shape of Functional Data

We explore different approaches for performing hypothesis tests on the shape of a mean function by developing general methodologies both, for the often assumed, i.i.d. error structure case, as well as for the more general case where the error terms have an arbitrary covariance structure. The procedures work by testing for patterns in the residuals after estimating the mean function and are extremely computationally fast. In the i.i.d. case, we fit a smooth function to the observed residuals and then fit similar functions to the permuted residuals. Under the null hypothesis that the curve comes from a particular functional shape, the permuted residuals should have a similar distribution to the unpermuted ones. So the fitted curves will have the same distribution thus allowing significance levels to be computed very efficiently. In the more general case, when several curves are observed, one can directly estimate the covariance structure and incorporate this into the analysis. However, when only one curve is observed, we adopt a graphical approach where one plots the p-value for differing levels of potential complexity in the covariance structure. This allows one to judge the degree of deviation from the assumed null distribution.We demonstrate the power of these methods on an extensive set of simulations.We also illustrate the approach on a data set of technology evolution curves which current theory suggests should have an underlying S shape. The developed techniques have wide potential applications in empirical testing of the shape of functional data