This paper derives new closed-form expressions for the masses of negative multinomial distributions. These masses can be maximized to determine the maximum likelihood estimator of its unknown parameters. An application to polarimetric image processing is investigated. We study the maximum likelihood estimators of the polarization degree of polarimetric images using different combinations of images. 1. Introduction The univariate negative binomial distribution is uniquely defined in many statistical textbooks. However, extensions defining multivariate negative multinomial distributions (NMDs) are more controversial. Most definitions are based on the probability generating function (PGF) of these distributions. Doss [1] proposed to define the PGF of an NMD as the inverse th power of a polynomial linear in each of its variables. This definition can also be found in the famous textbook [2, page 93] and the computation of its modes has been investigated in [3]. A more general class of NMDs introduced in [4] was characterized by PGFs of the form , where , is an matrix, and . In particular, matrices yielding infinitely divisible PGFs were derived. Finally, Bar-Lev et al. [5] introduced NMDs whose PGFs are defined as the inverse th power of any affine polynomial. Necessary and sufficient conditions on the coefficients of this affine polynomial were derived to obtain the PGF of a multivariate distribution defined on (where is the set of nonnegative integers) [6]. These very general multivariate NMDs were recently used for image processing applications in [7]. The family of NMDs introduced in [5] can be defined as follows. Let us denote the set of the first nonzero integers. We denote as the monomial obtained by multiplying all the entries of the vector whose indexes belong to , where stands for any subset of the indexes. Let be an affine polynomial with respect to the variables such that . The NMD distribution defined at pair is represented by its PFG which is given by Such laws are denoted as . However, as explained in [6], all couples do not provide a valid NMD. More specifically, Bernardoff has derived a finite number of conditions over such that is the PGF of an NMD for all positive integer . The corresponding expression of the coefficient of in the Taylor expansion of is given by the formula where and is the set of all subsets of . However, this expression of does not allow us to explicitly compute the masses of NMDs in the general case. As a first goal of this paper, we propose a way of computing the masses of multivariate NMDs defined above. A specific
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