Families of Discrete Distributions Satisfying Taylor's Power Law

Abstract

Count data in entomological studies can often be described by some form of contagious distribution, such as the negative binomial, Neyman type A, or Polya-Aeppli. These are all clustered Poisson distributions (their distributional forms are given in Section 2), arising when groups of individuals occur at random (i.e., are Poissonian) and individuals within a group have their own distribution. Whilst Pahl (1969) found strong evidence that the negative binomial distribution gives good fits to his 35 samples of data on the chrysomelid beetle Paropsis atomaria, Martin and Katti (1965) fitted a number of distributions to a variety of insect, plant, and animal data, and observed, for instance, that McGuire, Brindley, and Bancroft's (1957) data on the European cornborer Pyrausta nubilalis and Beall's (1940) data on the beet webworm Loxostege sticticalis are fitted well not only by the negative binomial, but also by the Neyman type A and other distributions. In such circumstances the fits given by different distributions are often very similar and the exact choice is not critical (see Kemp and Kemp, 1965). The recent work of Tripathi, Gurland, and Bhalerao (1986) gives further guidance on the choice of appropriate distribution when this is not clearly indicated by the ecological context. The negative binomial, Neyman type A, and Polya-Aeppli distributions all have two parameters. When there are many samples of data in one data set; Anscombe (1949) and Bliss and Owen (1958) suggested fitting a model that is parsimonious in parameters, by keeping one of them constant from sample to sample. Their use of a common exponent parameter, k, when fitting families of negative binomials was criticised by Taylor, Woiwod, and Perry (1979), who considered that k is an unstable parameter whose relationship with aggregation is doubtful. Much empirical evidence has accumulated in support of Taylor's power law