  Models such as the zero-inflated and zero-altered Poisson and zero-truncated
binomial are well-established in modern regression analysis. We propose a super
model that jointly and maximally unifies alteration, inflation, truncation and
deflation for counts, given a 1- or 2-parameter parent (base) distribution.
Seven disjoint sets of special value types are accommodated because all but
truncation have parametric and nonparametric variants. Some highlights include:
(i) the mixture distribution is exceeding flexible, e.g., 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) overdispersion can be studied
holistically in terms of the four operators; (iv) an important application:
heaped and seeped data from retrospective self-reported surveys are readily
handled, e.g., spikes and dips which are located virtually anywhere; (v) 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; (vi) the
VGAM R package implements the methodology based on Fisher scoring and
multinomial logit model (Poisson, NB, zeta and logarithmic parents are
implemented.) The GAITD-NB has potential to become the Swiss army knife of
count distributions.
