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.