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The generalized negative binomial distribution (GNB) is a new flexible familyof discrete distributions that are mixed Poisson laws with the mixinggeneralized gamma (GG) distributions. This family of discrete distributions isvery wide and embraces Poisson distributions, negative binomial distributions,Sichel distributions, Weibull--Poisson distributions and many other types ofdistributions supplying descriptive statistics with many flexible models. Thesedistributions seem to be very promising for the statistical description of manyreal phenomena. GG distributions are widely applied in signal and imageprocessing and other practical problems. The statistical estimation of theparameters of GNB and GG distributions is quite complicated. To find estimates,the methods of moments or maximum likelihood can be used as well as two-stagegrid EM-algorithms. The paper presents a methodology based on the search forthe best distribution using the minimization of $\ell^p$-distances and$L^p$-metrics for GNB and GG distributions, respectively. This approach, first,allows to obtain parameter estimates without using grid methods and solvingsystems of nonlinear equations and, second, yields not point estimates as themethods of moments or maximum likelihood do, but the estimate for the densityfunction. In other words, within this approach the set of decisions is not aEuclidean space, but a functional space.

A functional approach to estimation of the parameters of generalized negative binomial and gamma distributions