Smooth tests of goodness of fit

Smooth tests of goodness of fit assess the fit of data to a given probability density function within a class of alternatives that differs ‘smoothly’ from the null model. These alternatives are characterized by their order: the greater the order the richer the class of alternatives. The order may be a specified constant, but data-driven methods use the data to select the order and give tests that are unlikely to miss important effects. When testing for distributions within exponential families the test statistic often has a very convenient form, being the sum of squares of components that are asymptotically independent and asymptotically standard normal. The number of components is strongly related to the order of the alternatives. If the data differ from the null model the components provide diagnostics, giving an indication of how they differ. Outside of exponential families generalized smooth tests incorporating a Cholesky decomposition are needed to obtain test statistics with the convenient form mentioned. Using the components to give moment diagnostics of the alternative is more problematic than thought initially. These may be augmented or replaced by model selection methods that give a smooth model and a graphical depiction of the real model