Informative statistical analyses using smooth goodness of fit tests

We propose a methodology for informative goodness of fit testing that combines the merits of both hypothesis testing and nonparametric density estimation. In particular, we construct a data-driven smooth test that selects the model using a weighted integrated squared error (WISE) loss function. When the null hypothesis is rejected, we suggest plotting the estimate of the selected model. This estimate is optimal in the sense that it minimises the WISE loss function. This procedure may be particularly helpful when the components of the smooth test are not diagnostic for detecting moment deviations. Although this approach relies mostly on existing theory of (generalised) smooth tests and nonparametric density estimation, there are a few issues that need to be resolved so as to make the procedure applicable to a large class of distributions. In particular, we will need an estimator of the variance of the smooth test components that is consistent in a large class of distributions for which the nuisance parameters are estimated by method of moments. This estimator may also be used to construct diagnostic component tests. The properties of the new variance estimator, the new diagnostic components and the proposed informative testing procedure are evaluated in several simulation studies. We demonstrate the new methods on testing for the logistic and extreme value distributions.