Goodness-of-fit tests for the one-sided L\'evy distribution based on quantile conditional moments

In this paper we introduce a novel statistical framework based on the first two quantile conditional moments that facilitates effective goodness-of-fit testing for one-sided L\'evy distributions. The scale-ratio framework introduced in this paper extends our previous results in which we have shown how to extract unique distribution features using conditional variance ratio for the generic class of {\alpha}-stable distributions. We show that the conditional moment-based goodness-of-fit statistics are a good alternative to other methods introduced in the literature tailored to the one-sided L\'evy distributions. The usefulness of our approach is verified using an empirical test power study. For completeness, we also derive the asymptotic distributions of the test statistics and show how to apply our framework to real data.