Independent Additive Weighted Bias Distributions and Associated Goodness-of-Fit Tests

We use a Stein identity to define a new class of distributions which we call “independent additive weighted bias distributions.” We investigate related \(L^2\) -type discrepancy measures, empirical versions of which not only encompass traditional ODE-based procedures but also offer novel methods for conducting goodness-of-fit tests in composite hypothesis testing problems. We determine critical values for these new procedures using a parametric bootstrap approach and evaluate their power through Monte Carlo simulations. As an illustration, we apply these procedures to examine the compatibility of two real datasets with a compound Poisson gamma distribution.