(23)Since X��C-�� one must have �� = ��N(p, ��N) ? ��Y(��N, ��, ��).4. Mean Orthogonal Characterization of the Compound Multiparameter Hermite GammaThe mean orthogonal characterization of the compound gamma selleck Tofacitinib distribution allows for a wide variety of count data distributions in the mean orthogonal class. In order to reduce further the possible set of count distributions that can be used, one can ask for characterizations in terms of additional assumptions. For example, Puig [25] and Puig and Valero [26] characterize count data distributions satisfying Gauss’s principle and several notions of additivity, which via Theorem 5 can be translated to characterizations of compound gamma distributions.

Based on a result by Puig and Valero [20], we derive a most stringent characterization, which allows compounding of the gamma distribution under a single count data family, namely, the multiparameter Hermite distribution. To show this, some additional preliminaries are required.Definition 10 ��Let N be a counting random variable, let B1(p), B2(p),��, be independent and identically distributed Bernoulli random variables with probability of success p (0,1], and let N be independent of Bi(p), i = 1,2,��. Then N(p) = ��i=1NBi(p)(X(p) = 0ifN = 0) is called an independent p-thinning of N.Definition 11 ��Let F be a family of count distributions. It is called closed under binomial subsampling if, for any random variable N with distribution in F, all its independent p-thinnings, for all p (0,1], have distributions in F.Definition 12 ��Let F be a family of distributions.

It is called closed under convolution if, for any two independent random variables X, Y with distributions in F, the distribution of the sum X + Y also belongs to F.Definition 13 ��Let N be an integer random variable with pgf P(s) and factorial cumulant generating function (fcgf) CNf(s) = ln P(s + 1). For any integer n �� 1 the nth factorial cumulant of N is defined and denoted by ��(n) = dnCNf(s)/dsn|s=0.There is only one count distribution family closed under convolution and binomial subsampling.Theorem 14 (Characterization of the multiparameter Hermite distribution) ��Let F be a family of count distributions parameterized by its r first factorial cumulants �� = (��(1),��, ��(r)) and assume that its pgf P(s) Anacetrapib is continuous in �� over its parameter space. Then F is closed under convolution and binomial subsampling if and only if the pgf is of the formP(s)=exp?��k=1r��(k)k!(s?1)k.(24)Proof ��See Puig and Valero [20], proof of Theorem 1. Some comments are in order. The case r = 1 corresponds to the Poisson distribution, r = 2 is the Hermite distribution (e.g., [27]). For arbitrary r, this distribution is called the multiparameter Hermite distribution of order r by Milne and Westcott [28].