Bootstrap tests for almost goodness-of-fit

We introduce the $\mathit{\text{almost}}\mathit{\text{goodness-of-fit}}$ test, a procedure to assess whether a (parametric) model provides a good representation of the probability distribution generating the observed sample. Specifically, given a distribution function $F$ and a parametric family $\mathcal{G}=\{G(\mathit{\theta }):\mathit{\theta }\in \mathrm{\Theta }\}$, we consider the testing problem $$H_0:\mathrm{\parallel }F-G({\mathit{\theta }}_F){\mathrm{\parallel }}_p\ge \epsilon \text{vs}{H}_1:\mathrm{\parallel }F-G({\mathit{\theta }}_F){\mathrm{\parallel }}_p<\epsilon ,$$ where $\epsilon >0$ is a margin of error and $G({\mathit{\theta }}_F)$ denotes a representative of $F$ within the parametric class. The approximate model is determined via an M-estimator of the parameters. %The objective is the approximate validation of a distribution or an entire parametric family up to a pre-specified threshold value. The methodology also quantifies the percentage improvement of the proposed model relative to a non-informative (constant) benchmark. The test statistic is the ${\mathrm{L}}^p$-distance between the empirical distribution function and that of the estimated model. We present two consistent, easy-to-implement, and flexible bootstrap schemes to carry out the test. The performance of the proposal is illustrated through simulation studies and analysis and real-data applications.