Dr. Vladimir Vinogradov

Department of Mathematics

Ohio University

* Title: *"
Generating Canonical Generalized Linear Models Starting from Members of

the Extended Family of Zero-modified Geometric Distributions -- a Case Study"

* Abstract:*

We construct a new class of the canonical generalized linear models for

counts starting from the members of the extended family of zero-modified

geometric distributions. Some representatives of this class provide an

alternative to zero-inflated negative binomial regression, whereas

numerous new properties of members of this family which we establish and

their significant popularity in applications ascertain that our class is

quite diverse and more attractive. We decompose this class of probability

laws into the union of the non-overlapping natural exponential families,

which correspond to different values of an invariant of the exponential

tilting. We derive a unified closed-form expression for the variance

function of all these families, that involves the same invariant. In every

case, this function belongs to the Babel class. Several closed-form

representations for the relevant probability functions and cumulants, and

a unified formula for Shannon entropy are given. For each family considered,

this function is "locally Poisson" at zero, whereas the locally gamma

behavior at infinity holds with the exception of the border, Bernoulli family.

We determine a critical point in the range of values of the invariant of

exponential tilting that separates our family into two distinct classes

characterized by different J\o rgensen sets. This phenomenon is closely

related to Letac's decomposition of the Babel class into castes. In the

case where the infinite divisibility does not hold, different members of

the corresponding model exhibit under-, equi- and overdispersion. We prove

that each natural exponential family comprised of zero-modified geometric

distributions is self-reciprocal. The location of the mode(s) for all the

members of this class are determined. We provide a closed-form representation

for the asymptotically unbiased and efficient MLE of the invariant of the

exponential tilting. We derive the orthogonality of this MLE to the sample

mean and also the asymptotic normality of their joint distribution by

determining the relevant Fisher information matrix. These results combined

with the delta-method yield such properties of MLE for the canonical link as

its asymptotic unbiasedness, efficiency and normality. The closed-form

expression for the deviance-type estimator of the scaling parameter is presented.