Gumbel-Reverse Gumbel (GRG) Model: A New Asymmetric IRT Model for Binary Data

We propose a novel asymmetric item response theory (AsymIRT) model based on a convex combination of the complementary log–log and log–log links. These two links are the cumulative distribution functions (CDFs) of the Gumbel-min and Gumbel-max extreme value distributions, respectively. The resulting Gumbel-Reverse Gumbel (GRG) mixture model has one additional parameter. We illustrate using intelligence data taken from the Synthetic Aperture Personality Assessment (SAPA). In particular, we illustrate how nonparametric bootstrapping can be used to study model identification. We conclude with a discussion of how the GRG model fits in the AsymIRT literature more broadly as well as a call for scholars to work on comparing the AsymIRT models to each other in a more rigorous manner, in particular focusing on identification issues within AsymIRT models that do not fix a tail direction a priori.