Generally Altered, Inflated, Truncated and Deflated Regression

Abstract

Models such as the zero-inflated and zero-altered Poisson and zero-truncated binomial are now well established. We review important ideas behind the three operators involved (as well as deflation and modification) and other incremental extensions. We propose a super mixture model that unifies alteration, inflation, truncation and deflation for counts, given a 1- or 2-parameter parent or base distribution. Since all of the operators except for truncation have both parametric and nonparametric variants, it is necessary to consider a total of seven different scenarios. Highlights of this paper include the following: (i) the mixture distribution is exceeding flexible, accommodating up to seven modes; (ii) under-, equi- and over-dispersion can be handled using a negative binomial (NB) parent, with underdispersion handled by a novel Generally-Truncated-Expansion method; (iii) under- and over-dispersion are studied holistically in terms of the the four operators previously referred to; (iv) while generally-altered regression explains why observations are there, generally-inflated regression accounts for why they are there in excess, and generally-deflated regression explains why observations are not there. The important application of heaped and seeped data from retrospective self-reported surveys is briefly mentioned. The GAITD-NB has potential to become a Swiss army knife for analyzing count responses.