A new set of tools for goodness-of-fit validation

Abstract

We introduce two new tools to assess the validity of statistical distributions. Both the simple and composite null hypothesis contexts are considered. These tools are based on components derived from a new statistical quantity, the comparison curve, which can provide a detailed appraisal of validity. The first tool is a graphical representation of these components on a bar plot (B-plot) accompagnied with related local acceptance regions. These allow getting some ideas and building some confidence about where and to which extent the data contradict the model. The knowledge such gained could also suggest an existing goodness-of-fit test to supplement this assessment with a control of the type I error. Otherwise, a new test may be preferable and the second tool is is the combination of these components to produce a powerful ${\mathit{\chi }}^2$-type goodness-of-fit test. Because the number of these components can be large, we introduce new selection rules to decide on their number. In simulations, our new adaptive goodness-of-fit tests are powerwise competitive with the best solutions recommended. Practical examples show how to use these tools to derive principled information regarding if and possibly where the model departs from the data.