This paper proposes multivariate versions of the continuous Lindley mixture of Poisson distributions considered by Sankaran (1970). This new class of distributions can be used for modelling multivariate dependent count data when marginal overdispersion is present. After discussing some of its properties, a general multivariate model with Poisson-Lindley marginals and with a flexible covariance structure is proposed. Several specific models as well as one that allows correlations of any sign are considered, and then some estimation methods are discussed. Finally, some illustrative examples are given for fitting and demonstrating the usefulness of these bivariate distributions.

Axon diameters of retinal ganglion cells were measured from electron micrographs of the nerve fiber layer of the cat. Three adult retinae were examined which had mean axonal diameters of 1.18 ± 0.86 (n = 5553), 1.12 ± 0.79 (n = 7265), and 1.47 ±1.11 μm (n = 10,867). Cumulative histograms from several locations adjacent to the optic disc were unimodal (modal peaks: 0.6–0.8 μm). This unimodal distribution, however, did not reflect the regional differences in axonal diameters found throughout the retina. In many locations, especially those related to axons of the temporal retina, axon diameter distributions were clearly bimodal or even trimodal (modal peaks: 0.6–0.8, 1.4–2.1, and 3.3 μm). Measurements from one retina indicated that the mean diameters of axons arising from the area centralis and visual streak (0.94 ± 0.63 and 0.98 ± 0.68, respectively) were not significantly different from each other; however, when compared to other areas around the optic disc, the percentage of fibers with diameters between 1.5–2.0 μm was highest in the sample adjacent to the area centralis. Axons temporal to the optic disc were found to be on average larger than those nasal to the optic disc; similarly superior axons were larger than inferior axons. Axonal distributions at the retinal periphery were found to be significantly different from those at the optic disc (p ≤ 0.05) and contained a higher percentage of medium-sized axons and fewer small axons. In each of the three retinae the proportions small, medium, and large axons were respectively γ: 46; 47; 48, β: 50; 49; 48, and α: 4; 4; 4; regional differences in the proportions of each axonal class are compared to previously published ganglion cell density maps. Differences between axonal bundles within each sample location were not significantly different; however, in one retina axons in the scleral half of the fiber layer were significantly larger (P ≤ 0.01) than axons in the vitreal half of the nerve fiber layer adjacent to the optic disc. When compared to the axonal diameter distributions found within the optic nerve (Cottee et al., 1991) and optic tract (Reese et al., 1991), our data indicates that the diameter of retinal axons may increase by up to 30% along the length of the visual pathway.

Observed field data from a range of geographically distinct human communities suggest a consistent non-linear relationship between prevalence and mean intensity of Ascaris lumbricoides infection. Utilizing the negative binomial distribution as a description of observed aggregation, maximum-likelihood analysis reveals that the degree of aggregation is a negative linear function of mean worm burden. The factors responsible for this relationship in human populations require further study but may involve some combination of (i) density-dependent reduction in worm numbers within individuals, (ii) density-dependent parasite-induced host mortality or (iii) self-treatment by heavily infected hosts. Variability in the degree of aggregation appears dependent on the level of infection in a community and independent of geographical differences in the host or parasite populations.

Prevalence of infection with intestinal helminths is usually assessed through stool examination. There are limitations in using this technique to measure infection status, since some infected individuals have no patent infection. The relationship between infection assessed by stool examination and actual infection prevalence is investigated for the three major nematode species-Ascaris lumbricoides, Trichuris trichiura and the bookworms-using a model which describes the presence of non-egg producing worm combinations. The analysis demonstrates that stool examination under-estimates the actual infection prevalence, and that the degree of under-estimation is dependent on the level of infection, the nematode species and the parasite sex ratio. These findings have implications for the validity of epidemiological surveys and the evaluation of control programmes.

The frequency distribution of first generation, Steinernema feltiae Filipjev parasitic stages was over-dispersed with the majority of hosts containing few or no parasitic stages, whilst a few hosts contained a great many. Because of high extraction efficiency, the frequency distributions of the parasitic stages and the infective stages in the soil were assumed to be directly related. To explain the frequency distribution of the parasites it was therefore necessary to account for the frequency distribution of the S. feltiae infective stages in the soil. The infective stages were spatially aggregated into 30 cm diameter patches at the site of host death. These patches were randomly distributed approximately 1 m apart. At the 1 m scale, the pooled counts of infective stages were randomly distributed. Thus, in contrast to the frequency distributions, the spatial structuring of S. feltiae changed with the spatial scale of the interaction. This dynamic spatial structuring means that the majority of samples taken would contain few or no infective stages, whilst a few soil samples would contain a great many. Thus, the spatial structuring of the infective stages generates the over-dispersed frequency distribution of the S. feltiae in the soil. Hosts, encountering infective stages from this spatial distribution will, therefore, show an over-dispersed frequency distribution of S. feltiae parasitic stages.

We apply Lemaire's algorithm and a non-parametric mixed Poisson fit to a motor insurance portfolio in order to find the true claim frequency and claim amount distributions. The algorithm we develop accounts for the fact that observed distributions are distorted by bonus hunger, when a bonus-malus system is used by the insurer.

The head capsule widths of larvae of the cowpea pod-borer Maruca testulalis, reared individually and crowded together in the laboratory were measured. The measurements were subjected to a test of analysis of variance, frequency distribution and Dyar's rule. All methods gave a total of five larval instars in the life cycle and conformed to Dyar's rule. There was a high positive correlation (r = 0.97) between the size of the head capsule and the period of development elapsing. The minimum duration of larval development was 9 days.