In this paper, we compare the entropy of the original distribution and its corresponding compound distribution. Several results are established based on convex order and relative log-concave order. The necessary and sufficient condition for a compound distribution to be log-concave is also discussed, including compound geometric distribution, compound negative binomial distribution and compound binomial distribution.

During an epidemic outbreak, typically only partial information about the outbreak is known. A common scenario is that the infection times of individuals are unknown, but individuals, on displaying symptoms, are identified as infectious and removed from the population. We study the distribution of the number of infectives given only the times of removals in a Markovian susceptible–infectious–removed (SIR) epidemic. Primary interest is in the initial stages of the epidemic process, where a branching (birth–death) process approximation is applicable. We show that the number of individuals alive in a time-inhomogeneous birth–death process at time $t \geq 0$, given only death times up to and including time t, is a mixture of negative binomial distributions, with the number of mixing components depending on the total number of deaths, and the mixing weights depending upon the inter-arrival times of the deaths. We further consider the extension to the case where some deaths are unobserved. We also discuss the application of the results to control measures and statistical inference.

This paper probabilistically explores a class of stationary count time series models built by superpositioning (or otherwise combining) independent copies of a binary stationary sequence of zeroes and ones. Superpositioning methods have proven useful in devising stationary count time series having prespecified marginal distributions. Here, basic properties of this model class are established and the idea is further developed. Specifically, stationary series with binomial, Poisson, negative binomial, discrete uniform, and multinomial marginal distributions are constructed; other marginal distributions are possible. Our primary goal is to derive the autocovariance function of the resulting series.

Intensive field surveys were conducted in eastern Nebraska to determine the frequency distribution model and associated parameters of broadleaf and grass weed seedling populations. The negative binomial distribution consistently fit the data over time (1992 to 1993) and space (fields) for both the inter and intrarow broadleaf and grass weed seedling populations. The other distributions tested (Poisson with zeros, Neyman type A, logarithmic with zeros, and Poisson-binomial) did not fit the data as consistently as the negative binomial distribution. Associated with the negative binomial distribution is a k parameter. k is a nonspatial aggregation parameter related to the variance at a given mean value. The k parameter of the negative binomial distribution was consistent across weed density for individual weed species in a given field except for foxtail spp. populations. Stability of the k parameter across field sites was assessed using the likelihood ratio test There was no stable or common k value across field sites and years for all weed species populations. The lack of stability in k across field sites is of concern, because this parameter is used extensively in the development of parametric sequential sampling procedures. Because k is not stable across field sites, k must be estimated at the time of sampling. Understanding the variability in it is critical to the development of parametric sequential sampling strategies and understanding the dynamics of weed species in the field.

A better understanding of the influence of various crop and weed management practices on spatiotemporal dynamics of weeds could improve the design of integrated weed management systems. We examined the influence of 18- and 76-cm soybean row spacings on emergence pattern and spatial aggregation of giant foxtail, common lambsquarters, and velvetleaf seedling cohorts. In addition, we characterized the soil seedbank and determined the quantitative and spatial relationship between the seedbank and seedling populations. Viable seeds of about 10 weed species and twice as many species of seedlings were identified in the weed community. Giant foxtail and common lambsquarters were the predominant species in the seedling and seedbank population, respectively, each accounting for 60 to 70% of the total weed species density. Emergence of giant foxtail, common lambsquarters, and velvetleaf depleted 12 to 33%, < 2% and 12 to 49% of the seedbank in the upper 10 cm of the soil profile. Peak time and periodicity of weed emergence was not influenced by soybean row spacing, and peak time of emergence of giant foxtail, common lambsquarters, and velvetleaf occurred 3 to 4, 3 to 6, and 3 to 9 weeks after soybean planting (WAP), respectively. Magnitude of giant foxtail emergence 5, 6, and 9 WAP was 98, 96, and 76% greater in 76- than in 18-cm row soybeans only when the population of 76-cm row soybeans was 57% lower than the 18-cm soybeans in 1997. Giant foxtail and common lambsquarters seeds in the seedbank were aggregated in 1996 and 1997 according to the Taylor power law (TPL) and the negative binomial distribution (NBD). The TPL and the NBD were similar in describing the spatial aggregation of giant foxtail and common lambsquarters but not some velvetleaf seedling cohorts. The spatial aggregation of seedlings varied among cohorts for different weed species and was likely due to species-specific biological characteristics that influence seed dispersal, germination, and seedling emergence. Within a 1.5-ha area, aggregation declined with decreasing density. Within a 24-m2 area, the level of aggregation of all weed species decreased as seedling densities increased. These results indicated that soybean row spacing influenced neither weed emergence pattern nor weed spatial aggregation; thus, several management decisions can be similar in 18- and 76-cm row soybeans.

Full-count random sampling has been the traditional method of obtaining weed densities. Currently it is the recommended scouting procedure when using HERB, a herbicide selection decision aid. However, alternative methods of scouting that are quicker and more economical need to be investigated. One possibility that has been considered is binomial sampling. Binomial sampling is the procedure by which density is estimated from the number of random quadrats in which the count of individuals is equal to or less than a specified cutoff value. This sampling method has been widely used for insect scouting. There has also been interest in using binomial sampling for weed scouting. However, an economic analysis of this sampling method for weeds has not been performed. In this paper, the results of an economic analysis using simulations with binomial sampling and the HERB model are presented. Full-count sampling was included in the simulations to provide a benchmark for comparison. The comparison was made in terms of economic losses incurred when the estimated weed density obtained from sampling was inaccurate and a herbicide treatment was selected that did not maximize profits. These types of losses are referred to as opportunity losses. The opportunity losses obtained from the simulations indicate that in some situations binomial sampling may be a viable economic alternative to full-count sampling for fields with weed populations that follow a negative binomial distribution, assuming no prior knowledge of weed densities or negative binomial k values.

Superposition is a mapping on point configurations that sends the n-tuple into the n-point configuration , counted with multiplicity. It is an additive set operation such that the superposition of a k-point configuration in is a kn-point configuration in . A Poisson superposition process is the superposition in of a Poisson process in the space of finite-length -valued sequences. From properties of Poisson processes as well as some algebraic properties of formal power series, we obtain an explicit expression for the Janossy measure of Poisson superposition processes, and we study their law under domain restriction. Examples of well-known Poisson superposition processes include compound Poisson, negative binomial, and permanental (boson) processes.

For a family of linear preferential attachment graphs, we provide rates of convergence for the total variation distance between the degree of a randomly chosen vertex and an appropriate power law distribution as the number of vertices tends to ∞. Our proof uses a new formulation of Stein's method for the negative binomial distribution, which stems from a distributional transformation that has the negative binomial distributions as the only fixed points.

Objective – To present three probabilistic models (Poisson, negative binomial, Waring) to analyze the distribution of the number of contacts of patients followed using a psychiatric case register. Design – Longitudinal to obtain the distribution of the number of contacts (during 91 days following that of the first contact) observed on patients followed using the South-Verona Psychiatric Case Register during the period 1/1/79-31/12/91. Results – There were a total number of 6913 contacts on 3454 subjects. The chi-square test for the goodness of fit yields a significant result both for the Poisson distribution (3580 with 6 degrees of freedom, p < 0.001) and for the negative binomial distribution (65.47 with 18 degrees of freedom, p < 0.001); on the other hand a non significant result was obtained for the Waring distribution (25.31 with 19 degrees of freedom, p = 0.151). Conclusions – The Poisson distribution gave a very poor fit for the distribution of contacts. The negative binomial distribution could be employed to analyze the pattern of contacts when the right tail of the distribution is not important. The Waring distribution is the best of the three presented. Moreover, the variance of the Waring distribution can be decomposed in three components: a random component, a component which accounts for endogenous factors and another component which accounts for esogenous factors. Therefore the Waring distribution is useful when one wants to make comparisons between psychiatric case registers of the same country or of different countries.

Melamed's theorem states that, for a Jackson queueing network, the equilibrium flow along a link follows a Poisson distribution if and only if no customers can travel along the link more than once. Barbour and Brown (1996) considered the Poisson approximate version of Melamed's theorem by allowing the customers a small probability p of travelling along the link more than once. In this note, we prove that the customer flow process is a Poisson cluster process and then establish a general approximate version of Melamed's theorem that accommodates all possible cases of 0 ≤ p < 1.

For several pairs (P, Q) of classical distributions on ℕ0, we show that their stochastic ordering P ≤stQ can be characterized by their extreme tail ordering equivalent to P({k*})/Q({k*}) ≥ 1 ≥ limk→k*P({k})/Q({k}), with k* and k* denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P({k*})/Q({k*}) for finite k*. This includes in particular all pairs where P and Q are both binomial (bn1,p1 ≤stbn2,p2 if and only if n1 ≤ n2 and (1 - p1)n1 ≥ (1 - p2)n2, or p1 = 0), both negative binomial (b−r1,p1 ≤stb−r2,p2 if and only if p1 ≥ p2 and p1r1 ≥ p2r2), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new. The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Lévy measures. We give four proofs in the binomial case (methods (i)-(iv)) and three in the negative binomial case (methods (i), (iv), and (v)). The statement for hypergeometric distributions is proved via method (i).

An earlier paper (May, 1977) developed a theoretical framework for exploring the consequences of dioecy for the population dynamics of schistosomes, assuming an unbiased sex ratio. This paper extends the analysis to biased sex ratios, as have been reported in practice. We consider the relationships of the mean number and distribution of worms among hosts, the sex ratio, and the mating system (monogamous or polygamous) to: (i) the female mating probability, Φ the prevalence of mated female worms. Ω: and (iii) the mean number of mated female worms per host, ξ. Among other results, we show how high values of Φ are associated with male-biased sex ratios and polygamous mating; that Ω is independent of the mating system and is relatively unaffected by the sex ratio; and that ξ is maximal for unbiased sex ratios given monogamous mating, and for female-biased sex ratios if mating is polygamous. These results, together with the confounding effects of the mean number and distribution of worms, are described in detail in the main body of the paper.

The characteristically aggregated frequency distribution of macroparasites in their hosts is a key feature of host–parasite population biology. We begin with a brief review of the theoretical literature concerning parasite aggregation. Though this work has illustrated much about both the sources and impact of parasite aggregation, there is still no definitive analysis of both these aspects. We then go on to illustrate the use of one approach to this problem – the construction of Moment Closure Equations (MCEs), which can be used to represent both the mean and second moments (variances and covariances) of the distribution of different parasite stages and phenomenological measures of host immunity. We apply these models to one of the best documented interactions involving free-living animal hosts – the interaction between trichostrongylid nematodes and ruminants. The analysis compares patterns of variability in experimental infections of Teladorsagia circumcincta in sheep with the equivalent wildlife situation – the epidemiology of T. circumcincta in a feral population of Soay sheep on St Kilda, Outer Hebrides. We focus on the relationship between mean parasite load and aggregation (inversely measured by the negative binomial parameter, k) for cohorts of hosts. The analysis and empirical data indicate that k tracks the increase and subsequent decline in the mean burden with host age. We discuss this result in terms of the degree of heterogeneity in the impact of host immunity or parasite-induced mortality required to shorten the tail of the parasite distribution (and therefore increase k) in older animals. The model is also used to analyse the relationship between estimated worm and egg counts (since only the latter are often available for wildlife hosts). Finally, we use these results to review directions for future work on the nature and impact of parasite aggregation.

We demonstrate, by truncating the expected negative binomial distribution, that the tracheal mite parasite, Acarapis woodi, causes mortality in the Europen honey bee, Apis mellifera, but, that this mortality can be documented only during periods of low mite densities (mites per bee). At high mite densities, this technique no longer reveals mite-induced mortality. We suggest that this paradox results from a reduction in the mortality threshold at high mite densities, concealing mortality from our truncation procedures. Longitudinal comparisons of hive and forager bee cohorts, nevertheless, suggests mortality at all levels of infestation. We propose that persistent parasitism overcomes a colony's ability to compensate for losses, leading to its sudden decline and death.

When detecting the adventitious presence of transgenic plants (AP), it is important to use an appropriate testing method in the laboratory. Dorfman's group testing method is effective for reducing the number of laboratory analyses, but does not consider the case where AP is diluted below the sensitivity of the analyses, which causes the rate of false negatives to increase. The objective of this study is to propose binomial and negative binomial probabilistic models for determining the required sample size (n), number of pools (g), and size of the pool (k) for detecting individuals possessing AP with a probability ≥ (1 − α) (for a small α) given: (1) pool size (k); (2) estimated proportion of individuals with AP in the population (p); (3) concentration of the trait of interest (AP) in individual seeds (w); and (4) detection limit of the test (c) (AP concentration in a pool below which it cannot be detected). The proposed models consider the different rates of false positives (δ) and false negatives (λ), and the assessment of consumer and producer risks. Results have shown that when using the negative binomial, a required sample size n can be determined that guarantees a high probability that m individuals or g pools containing AP will be found. The pools formed have an optimum size, such that one element with AP will be detected at a low cost. The negative binomial distribution should be used when it is known that the proportion of individuals with AP in the population is p < 0.1; thus, it is guaranteed that m individuals or g pools of individuals with AP will be detected with high probability.

A Bayesian geostatistical model was developed to predict the intensity of infection with Schistosoma mansoni in East Africa. Epidemiological data from purpose-designed and standardized surveys were available for 31458 schoolchildren (90% aged between 6 and 16 years) from 459 locations across the region and used in combination with remote sensing environmental data to identify factors associated with spatial variation in infection patterns. The geostatistical model explicitly takes into account the highly aggregated distribution of parasite distributions by fitting a negative binomial distribution to the data and accounts for spatial correlation. Results identify the role of environmental risk factors in explaining geographical heterogeneity in infection intensity and show how these factors can be used to develop a predictive map. Such a map has important implications for schisosomiasis control programmes in the region.

Macroparasites are almost always aggregated across their host populations, hence the Negative Binomial Distribution (NBD) with its exponent parameter k is widely used for modelling, quantifying or analysing parasite distributions. However, many studies have pointed out some drawbacks in the use of the NBD, with respect to the sensitivity of k to the mean number of parasites per host or the under-representation of the heavily infected hosts in the estimate of k. In this study, we compare the fit of the NBD with 4 other widely used distributions on observed parasitic gastrointestinal nematode distributions in their sheep host populations (11 datasets). Distributions were fitted to observed data using maximum likelihood estimator and the best fits were selected using the Akaike's Information Criterion (AIC). A simulation study was also conducted in order to assess the possible bias in parameter estimations especially in the case of small sample sizes. We found that the NBD is seldom the best fit for gastrointestinal nematode distributions. The Weibull distribution was clearly more appropriate over a very wide range of degrees of aggregation, mainly because it was more flexible in fitting the heavily infected hosts. Moreover, the Weibull distribution estimates are less sensitive to sample size. Thus, when possible, we suggest to carefully check on observed data if the NBD is appropriate before conducting any further analysis on parasite distributions.

In this paper we address the question of detecting immunity to helminth infections from patterns of infection in endemic communities. We use stochastic simulations to investigate whether it would be possible to detect patterns predicted by theoretical models, using typical field data. Thus, our technique is to simulate a theoretical model, to generate the data that would be obtained in field surveys and then to analyse these data using methods usually employed for field data. The general behaviour of the model, and in particular the levels of variability of egg counts predicted, show that the model is capturing most of the variability present in field data. However, analysis of the data in detail suggests that detection of immunity patterns in real data may be very difficult even if the underlying patterns are present. Analysis of a real data set does show patterns consistent with acquired immunity and the implications of this are discussed.

This paper describes the distribution of Ascaris suum in experimentally and naturally infected pigs, and offers a comparison with A. lumbricoides infections in humans. In the first study, worms were recovered post-mortem from a group of 38 pigs that had been trickle inoculated with 10000 infective A. suum eggs twice weekly for 12 weeks. In the second study, worms were collected from a group of 49 pigs that had been kept on a pasture contaminated with infective A. suum eggs for 10 weeks, after which they received treatment with an anthelmintic; they then were turned out on the same pasture for a second 10-week period before slaughter. The worm burdens of the naturally infected pigs were recorded both at treatment and post-mortem. Mean worm counts were similar at all occasions but the prevalence of infection was higher in the trickle infected and naturally reinfected pigs. Furthermore, the prevalence in naturally infected pigs increased significantly over the study period. Worm burden distributions in all groups were heavily overdispersed, but the distribution patterns differed significantly between groups: lower exposure (initial natural infection) gave a low prevalence and an almost uniform distribution of worm burdens among infected hosts. Continued or higher exposure (trickle and natural reinfection) resulted in increased prevalence and a reduction in the proportion of hosts with increasing worm load. A positive correlation was found between initial and reinfection worm burdens in the naturally infected pig population, suggesting that individual pigs are predisposed to a high or low intensity of infection. The prevalence and intensity as well as the distribution observed for A. suum infection in pigs were comparable to those reported for A. lumbricoides in endemic areas, and there is evidence for predisposition to A. suum in pigs, with an estimated correlation coefficient similar to that found in humans. It is concluded that A. suum infections in pigs are a suitable model to study the population dynamics of A. lumbricoides in human populations.

We investigate the stationarity of minification processes when the marginal is a discrete distribution. There is a close relationship between the problem considered by Arnold and Isaacson (1976) and the stationarity in minification processes. We give a necessary and sufficient condition for a discrete distribution to be the marginal of a stationary minification process. Members of the Poisson and negative binomial families can be the marginals of stationary minification processes. The geometric minification process is studied in detail, and two characterizations of it based on the structure of the innovation process are given.