Almost goodness-of-fit tests

We introduce the almost goodness-of-fit test, a procedure to decide if a (parametric) model provides a good representation of the probability distribution generating the observed sample. We consider the approximate model determined by an M-estimator of the parameters as the best representative of the unknown distribution within the parametric class. The objective is the approximate validation of a distribution or an entire parametric family up to a pre-specified threshold value, the margin of error. The methodology also allows quantifying the percentage improvement of the proposed model compared to a non-informative (constant) one. The test statistic is the $\mathrm{L}^p$-distance between the empirical distribution function and the corresponding one of the estimated (parametric) model. The value of the parameter $p$ allows modulating the impact of the tails of the distribution in the validation of the model. By deriving the asymptotic distribution of the test statistic, as well as proving the consistency of its bootstrap approximation, we present an easy-to-implement and flexible method. The performance of the proposal is illustrated with a simulation study and the analysis of a real dataset.