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This chapter defines the COM–Poisson distribution in greater detail, discussing its associated attributes and computing tools available for analysis. This chapter first details how the COM–Poisson distribution was derived, and then describes the probability distribution, and introduces computing functions available in R that can be used to determine various probabilistic quantities of interest, including the normalizing constant, probability and cumulative distribution functions, random number generation, mean, and variance. The chapter then outlines the distributional and statistical properties associated with this model, and discusses parameter estimation and statistical inference associated with the COM–Poisson model. Various processes for generating random data are then discussed, along with associated available R computing tools. Continued discussion provides reparametrizations of the density function that serve as alternative forms for statistical analyses and model development, considers the COM–Poisson as a weighted Poisson distribution, and details discussion describing the various ways to approximate the COM–Poisson normalizing function.
This chapter is an overview summarizing relevant established and well-studied distributions for count data that motivate consideration of the Conway–Maxwell–Poisson distribution. Each of the discussed models provides an improved flexibility and computational ability for analyzing count data, yet associated restrictions help readers to appreciate the need for and usefulness of the Conway–Maxwell–Poisson distribution, thus resulting in an explosion of research relating to this model. For completeness of discussion, each of these sections includes discussion of the relevant R packages and their contained functionality to serve as a starting point for forthcoming discussions throughout subsequent chapters. Along with the R discussion, illustrative examples aid readers in understanding distribution qualities and related statistical computational output. This background provides insights regarding the real implications of apparent data dispersion in count data models, and the need to properly address it.
The Conway–Maxwell–Poisson distribution has garnered interest in and development of other flexible alternatives to classical distributions. This chapter introduces various distributional extensions and generalities motivated by functions of COM–Poisson random variables, including Conway–Maxwell-inspired generalizations of the Skellam distribution, binomial distribution, negative binomial distribution, the Katz class of distributions, two flexible series system life length distributions, and generalizations of the negative hypergeometric distribution.
While the Poisson distribution is a classical statistical model for count data, the distributional model hinges on the constraining property that its mean equal its variance. This text instead introduces the Conway-Maxwell-Poisson distribution and motivates its use in developing flexible statistical methods based on its distributional form. This two-parameter model not only contains the Poisson distribution as a special case but, in its ability to account for data over- or under-dispersion, encompasses both the geometric and Bernoulli distributions. The resulting statistical methods serve in a multitude of ways, from an exploratory data analysis tool, to a flexible modeling impetus for varied statistical methods involving count data. The first comprehensive reference on the subject, this text contains numerous illustrative examples demonstrating R code and output. It is essential reading for academics in statistics and data science, as well as quantitative researchers and data analysts in economics, biostatistics and other applied disciplines.
Many seed quality tests are conducted by first randomly assigning seeds into replicates of a given size. The replicate results are then used to check whether or not any problems occur in the realization of the test. The two main tools developed for this verification are the ratio of the observed variance of the replicate results to a theoretical variance and the tolerance for the range of the results. In this paper, we derive the theoretical distribution and its related properties of the sequence of numbers of seeds with a given quality attribute present in the replicates. From these theoretical results, we revisit the two quality checking tools widely used for the germination test. We show a precaution to be taken when relying on the variance ratio to check for under- or over-dispersion of the replicate results. This has led to the development of tables providing credible intervals of the variance ratio. The International Seed Testing Association tolerance tables for the range of the results are also compared with tolerances computed from the exact theoretical distribution of the range, leading us to recommend a revision of these tables.
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