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A multivariate Poisson distribution is a natural choice for modeling count data stemming from correlated random variables; however, it is limited by the underlying univariate model assumption that the data are equi-dispersed. Alternative models include a multivariate negative binomial and a multivariate generalized Poisson distribution, which themselves suffer from analogous limitations as described in Chapter 1. While the aforementioned distributions motivate the need to instead consider a multivariate analog of the univariate COM–Poisson, such model development varies in order to take into account (or results in) certain distributional qualities. This chapter summarizes such efforts where, for each approach, readers will first learn about any bivariate COM–Poisson distribution formulations, followed by any multivariate analogs. Accordingly, because these models are multidimensional generalizations of the univariate COM–Poisson, they each contain their analogous forms of the Poisson, Bernoulli, and geometric distributions as special cases. The methods discussed in this chapter are the trivariate reduction, compounding, Sarmanov family of distributions, and copulas.
The caridean shrimp family Chlorotocellidae has so far not been reported from Korean waters. One of the four genera in the family is herein reported, based on the species Anachlorocurtis commensalis. Specimens were collected from antipatharian black corals by scuba diving at depths ranging from 20–60 m from three localities: Jejudo Island (33°13′N) between the South Sea of Korea and the north-eastern East China Sea, Namhyeongjedo Islet (34°53′N) in the Korea Strait, and Dokdo Island (37°15′N) in the East Sea. The species had previously been reported in tropical to subtropical latitudes in the western Pacific, with known records from southern Taiwan (up to 21°55′N) to central Japan (down to 34°50′N). The species is thus recorded for the first time from a temperate region in the western Pacific Ocean, postulated to be influenced by a branch of the Kuroshio Warm Current.
Chapter 3 describes models for multivariate distributions. Included are the multinormal distribution, the multi-lognormal distribution, multivariate distributions constructed as products of conditional distributions, and three families of multivariate distributions with prescribed marginals and covariances: the Nataf family, the Morgenstern family, and copula distributions. Several structural reliability methods require transformation of original random variables into statistically independent standard normal random variables. The conditions for such a transformation to exist and be reversible are described and formulations are presented for each of the described multivariate distribution models. Also developed are the Jacobians of each transform and its inverse, which are also used in reliability analysis. Analytical and numerical results to facilitate the use of Nataf and Morgenstern distributions are provided in this chapter.
A simple epidemic process in which the number of individuals who can become infected at any point in time is itself a random variable is described. The discrete asymptotic behaviour of such a process is discussed. In particular, the associated marginal distribution of the limiting process is considered.
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