The Negative Binomial Distribution of Order k and Some of Its Properties

The negative binomial distribution of order k is introduced and briefly studied. First it is shown that it is a proper probability distribution. Then its probability generating function, mean and variance are derived. Finally it is shown that the number of trials until the rth kth consecutive success (r ≧ 1, k ≧ 1) in independent trials with constant success probability p (0 < p < 1) is distributed as negative binomial distribution of order k. The present paper generalizes results of SHANE (1973), PHILIPPOU and MUWAFI (1982), and PHILIPPOU, GEORGHIOU and PHILIPPOU (1982).     

On Characterizing Some Discrete Distributions by a Conditional Distribution and a Regression Function

In this paper we consider characterizations of the binomial, negative binomial, hypergeometric, negative hypergeometric, multinomial and multivariate hypergeometric distributions, by linear regression of one random variable (vector) on the other and the conditional distribution of the other random variable (vector) given the first. It is also indicated how these results can be used in genetics.     

On Mohan's Property of the Poisson Distribution

Two interesting results encountered in the literature concerning the Poisson and the negative binomial distributions are due to Moran (1952) and Patil & Seshadri (1964), respectively. Morans result provided a fundamental property of the Poisson distribution. Roughly speaking, he has shown that if Y, Z are independent, non-negative, integer-valued random variables with X = Y | Z then, under some mild restrictions, the conditional distribution of Y | X is binomial if and only if Y, Z are Poisson random variables. Motivated by Morans result Patil & Seshadri obtained a general characterization. A special case of this characterization suggests that, with conditions similar to those imposed by Moran, Y | X is negative hypergeometric if and only if Y, Z are negative binomials. In this paper we examine the results of Moran and Patil & Seshadri in the case where the conditional distribution of Y | X is truncated at an arbitrary point k – 1 (k = 1, 2, …). In fact we attempt to answer the question as to whether Morans property of the Poisson distribution, and subsequently Patil & Seshadris property of the negative binomial distribution, can be extended, in one form or another, to the case where Y | X is binomial truncated at k – 1 and negative hypergeometric truncated at k – 1 respectively.     

A comparative analysis on the distribution of nymphal populations of some leaf- and planthoppers on rice plant

The frequency distribution of the number of nymphs per hill of rice plant were analyzed for three species of rice leaf- and planthoppers, Nilaparvata lugens, Delphacodes striatella and Nephotettix cincticeps, based on the sampling data obtained during their last generations in the paddy field. For every species concerned, individual distributions were proved to be contagious and to fit well to the negative binomial distribution. Further, it was found that the value of negative binomial parameter k is so stable for same species that a single value of k is applicable for a series of counts with different means, whereas that k differs remarkably among different species: if the reciprocal of the weighted estimate of common k which is an adequate index measuring degree of contagiousness of the distribution, is compared among different species, it is higher in the order of Nilaparvata, Delphacodes and Nephotettix. The ecological and practical implication of constancy and heterogeneity within and among species was discussed respectively as to the value of parameter k.     

The spatial distribution of the wheat-bulb fly,Leptohylemyia coarctata (fall.) (Diptera: Anthomyiidae)

The spatial distribution of the eggs, larvae, pupae and adults of the wheat-bulb fly was investigated by fitting 42 sets of data comprising 1334 samples to the Poisson and negative binomial distributions, and by using the power law (S²=am^b). In general, the tests indicated that all stages were aggregated and fitted the negative binomial model.     

Influence of parental density on the distribution pattern of eggs in the common cabbage butterfly,Pieris rapae crucivora

To investigate the relation between the distribution pattern of eggs and the parental density in the common cabbage butterfly, Pieris rapae crucivora, the countings of egg number per plant were made on both cabbage plants cultivated in the farm and planted in the net house in which the female butterflies were released at various densities. The frequency distribution of eggs fits well to the negative binomial excepting the cases where they agree withPoisson series, and the degree of aggregation expressed as the reciprocal of the parameter, 1/k, tends to decrease as the egg or parental density increases. At the same parental density, however, the distribution of eggs can be described by the negative binomial with a common parameter, k_c, regardless of the difference in the density of laid eggs. In the case where a single female butterfly lays eggs, the spatial pattern of egg distribution is always lean, while its frequencies conform toPoisson or the negative binomial series. This lean changes toward patchy with increasing the parental density. From these results, it is concluded that the degree of aggregation in the distribution of eggs decreases with the increase of the parental density.     

Experimental studies on the distribution of the aphid counts

The present paper dealt with the sequential changes of the distribution pattern of apterous females aphid populations, that were artificially settled at the beginning on the experimental barley ‘field'. The aphids were settled at random or even with a fixed denisty per plant. For five or six days after the settling, the number of individuals followed the negative binomial distributions in all cases while the parameters k and p were varying. The estimated values of k were rather small for the first one week after the settling, which may suggest that the number of moving aphids between plants was relatively small and the degree of concentration expressing the intrinsic increase was high. After that, as the number of individuals increased, the number of moving aphids between plants would be considered to be increased. It was found that with the lapse of time the degree of concentration decreased or k became larger. The distribution of aphids per blade in a plant was also described briefly.     

Cumulative Damage Survival Models for the Randomized Complete Block Design

A continuous time discrete state cumulative damage process {X(t), t ≥ 0} is considered, based on a non‐homogeneous Poisson hit‐count process and discrete distribution of damage per hit, which can be negative binomial, Neyman type A, Polya‐Aeppli or Lagrangian Poisson. Intensity functions considered for the Poisson process comprise a flexible three‐parameter family. The survival function is S(t) = P(X(t) ≤ L) where L is fixed. Individual variation is accounted for within the construction for the initial damage distribution {P(X(0) = x) | x = 0, 1, …,}. This distribution has an essential cut‐off before x = L and the distribution of L – X(0) may be considered a tolerance distribution. A multivariate extension appropriate for the randomized complete block design is developed by constructing dependence in the initial damage distributions. Our multivariate model is applied (via maximum likelihood) to litter‐matched tumorigenesis data for rats. The litter effect accounts for 5.9 percent of the variance of the individual effect. Cumulative damage hazard functions are compared to nonparametric hazard functions and to hazard functions obtained from the PVF‐Weibull frailty model. The cumulative damage model has greater dimensionality for interpretation compared to other models, owing principally to the intensity function part of the model.     

Multinomial N-mixture models for removal sampling

Multinomial $N$-mixture models are commonly used to fit data from a removal sampling protocol. If the mixing distribution is negative binomial, the distribution of the counts does not appear to have been identified, and practitioners approximate the requisite likelihood by placing an upper bound on the embedded infinite sum. In this paper, the distribution which underpins the multinomial $N$-mixture model with a negative binomial mixing distribution is shown to belong to the broad class of multivariate negative binomial distributions. Specifically, the likelihood can be expressed in closed form as the product of conditional and marginal likelihoods and the information matrix shown to be block diagonal. As a consequence, the nature of the maximum likelihood estimates of the unknown parameters and their attendant standard errors can be examined and tests of the hypothesis of the Poisson against the negative binomial mixing distribution formulated. In addition, appropriate multinomial $N$-mixture models for data sets which include zero site totals can also be constructed. Two illustrative examples are provided.     

Oviposition pattern of phytophagous insects: on the importance of host population heterogeneity

Frequency distributions of insect immatures per host are often fitted to contagious distributions, such as the negative binomial, to deduce oviposition pattern. However, different mechanisms can be involved for each theoretical distribution and additional biological information is needed to correctly interpret the fits. We chose the chestnut weevil Curculio elephas, a pest of the European chestnut Castanea sativa, as a model to illustrate the difficulties of inferring oviposition pattern from fits to theoretical distributions and from the variance/mean ratio. From field studies over 13–16 years, we show that 20 out of the 31 yearly distributions available fit a negative binomial and 25 a zero-inflated Poisson (ZIP). No distribution fits a Poisson distribution. The ZIP distribution assumes heterogeneity within the fruit population. There are two categories of host: the first comprises chestnuts unsuitable for weevil oviposition or in excess relative to the number of weevil females, and the second comprises suitable fruits in which oviposition behavior is random. Our results confirm this host heterogeneity. According to the ZIP distribution, the first category of hosts includes on average 74% of the chestnuts. A negative binomial distribution may be generated by either true or false contagion. We show that neither interference between weevil females, nor spatial variation in the infestation rate exist. Consequently, the observed distributions of immatures are not the result of false contagion. Nevertheless, we cannot totally exlude true contagion of immatures. In this paper we discuss the difficulty of testing true contagion in natural conditions. These results show that we cannot systematically conclude in favour of contagion when fitting a distribution such as the negative binomial or when a variance/mean ratio is higher than unity. Received: 22 September 1997 / Accepted: 15 December 1997     

Seedling dynamics of <Emphasis Type="Italic">Acer mono</Emphasis> and <Emphasis Type="Italic">Fagus crenata</Emphasis>: an environmental filter limiting their adult distributions

Spatial pattern of a biotic population may change over time as its component individuals grow or die out, but whether this is the case for desert annual plants is largely unknown. Here we examined seasonal changes in spatial patterns of two annuals, Eriogonum abertianum and Haplopappus gracilis, in initial (winter) and final (summer) densities. The density was measured as the number of individuals from 384 permanent quadrats (each 0.5 m × 0.5 m) in the Chihuahuan Desert near Portal, Arizona, USA. We used three probability distributions (binomial, Poisson, and negative binomial or NB) that represent three basic spatial patterns (regular, random, and clumped) to fit the observed frequency distributions of densities of the two annuals. Both species showed clear clumped patterns as characterized by the NB and had similar inverse J-shaped frequency distribution curves in two density categories. Also, both species displayed a reduced degree of aggregation from winter to summer after the spring drought (massive die-off), as indicated by the increased k-parameter of the NB and decreased values of another NB parameter p, variance/mean ratio, Lloyd’s Index of Patchiness, and David and Moore’s Index of Clumping. Further, we hypothesized that while the NB (i.e., Poisson-logarithmic) well fits the distribution of individuals per quadrat, its components, the Poisson and logarithmic, may describe the distributions of clumps per quadrat and of individuals per clump, respectively. We thus obtained the means and variances for (1) individuals per quadrat, (2) clumps per quadrat, and (3) individuals per clump. The results showed that the decrease of the density from winter to summer for each plant resulted from the decrease of individuals per clump, rather than from the decrease of clumps per quadrat. The great similarities between the two annuals indicate that our observed temporal changes in spatial patterns may be common among desert annual plants.     

I σ-Index,a measure of dispersion of individuals

Conclusion and summary As theI _σ-index is neither affected by the mean per sample unit except for regular distribution nor standing on the assumption of any definite type of contagious distribution, it may have the most wide range of application among the ones hitherto deviced for measuring the dispersion of individuals in a population. The relations of theI _σ-index to ,k of negative binomial distribution andN of binomial distribution as well as the new dispersion index,I _B , given in this paper may serve, if necessary, for the analysis of data in the ecological works. Aided by the Scientific Research Fund from the Ministry of Education.     

Mixtures of Distributions by the Poisson Distribution of Order K

The Poisson-binomial, the Poisson-negative binomial (or Pascal) and Neyman's Type A distribution, which are Poisson mixtures of the binomial, the negative binomial and the Poisson distribution, respectively, have received a lot of attention in statistical literature [see e.g. Katti and Gurland (1961, 1962), Anscombe (1950), and Neyman (1939)]. In the present paper, their respective generalizations are introduced and briefly studied, when the Poisson distribution of order k [see Philippou (1983), Philippou, Georghiou and Philippou (1983), and Charalambides (1986)] replaces the Poisson distribution in its mixing role.     

Symmetric Bernoulli Distributions and Generalized Binomial Distributions

The generalized binomial distribution is defined as the distribution of a sum of symmetrically distributed Bernoulli random variates. Several two-parameter families of generalized binomial distributions have received attention in the literature, including the Polya urn model, the correlated binomial model and the latent variable model. Some properties and limitations of the three distributions are described. An algorithm for maximum likelihood estimation for two-parameter generalized binomial distributions is proposed. The Polya urn model and the latent variable model were found to provide good fits to sub-binomial data given by Parkes. An extension of the latent variable model to incorporate heterogeneous response probabilities is discussed.     

Spatial distribution of the green peach aphid used in estimating the populations of gynoparae

Summary In 2 years, during the initial invasion of peach leaves by the green peach aphid,Myzus persicae (Sulzer), the number of gynoparae was low, and the distribution on leaves was random. Then as the mean number increased, the distribution became intermediate and could not be distinguished from either a Poisson or a negative binomial. Finally, as the mean continued to increase, the variance increased rapidly, and the population was found to fit a negative binomial distribution. Thus the aggregation response was verified because the dispersion pattern fitted a contagious distribution. A sampling plan was devised by which the dispersion parameterk was used to estimate the density of aphids per leaf based on the percentage of leaves infested. Sampling the third year of the study confirmed the validity of the sampling parameter that had been calculated from data for the 2 previous years.     

On a restricted occupancy model and its applications

With the multivariate hypergeometric distribution as a background certain occupancy distributions useful in practical applications are derived. More specifically it is assumed that a sample of n individuals is drawn from a population consisting of m types with r individuals in each type, (i) without replacement and (ii) by returning the selected individual in the population and with it another individual of the same type. The distributions of the number Z of distinct types observed in the sample are obtained in both cases in terms of the numbers. Assuming, in addition to the m equiprobable types of individuals, the existence of a control type, say, with s individuals, the joint distribution of the number U of distinct types observed in the sample and the number V of individuals of the control type present in the sample is obtained in terms of the numbers C(n, k, r) and the marginal distribution of U in terms of the Gould-Hopper numbers. Using these distributions minimum variance unbiased estimators of the number m of types are derived. Moreover small sample tests based on the zero frequency are constructed.     

Effects of contagious distributions of parasitoid eggs per host and of sampling vagaries on Nicholson’s area of discovery

Summary Hassell andVarley described a negative relationship between the Nicholsonianarea of discovery and parasitoid density that permits coexistence of two or more parasitoid species and yields stability. They ascribed the relationship to “mutual interference” among searching parasitoid adults, leading to a “decreased efficiency”. This matter has recently been studied in more detail byHassell and coworkers. Negative relationships can occur for several reasons. In the present paper it is shown that the negative relation occurs under certain assumptions about the way parasitoid eggs are distributed among hosts. Several “contagious” (or “clustered”) distributions were studied. Those assuming variation in host susceptibility or accessibility (negative binomial and added zeros distributions) yielded negative relations. Those assuming variation in parasitoid aggressiveness and fecundity (Neyman Type A distributions) yielded variable results depending on how the distributions of parasitoids per host and of eggs per parasitoid were combined. It was found that, when analyzing actual data, sampling fluctuations can introduce strong negative relations either with or without “contagion”. In all instances, the relation betweenarea of discovery and parasitoid density is affected by host density and is, in general, not linear. The foregoing findings raise serious questions about thearea of discovery concept. An alternative approach is suggested. The initial phase of this study was supported by National Institute of Health Grant S-Rol-AI0611, under Prof.C. B. Huffaker, Division of Biological Control, Department of Entomology, University of California, Berkeley. The study was completed under National Science Foundation Grant GB38271X. Paper no. 4705 of the Journal Series of the North Carolina Agricultural Experiment Station Raleigh.     

Indirect evidence of density‐dependent population regulation in Aponomma hydrosauri (Acari: Ixodidae), an ectoparasite of reptiles

Abstract The extent to which density‐dependent processes regulate natural populations is the subject of an ongoing debate. We contribute evidence to this debate showing that density‐dependent processes influence the population dynamics of the ectoparasite Aponomma hydrosauri (Acari: Ixodidae), a tick species that infests reptiles in Australia. The first piece of evidence comes from an unusually long‐term dataset on the distribution of ticks among individual hosts. If density‐dependent processes are influencing either host mortality or vital rates of the parasite population, and those distributions can be approximated with negative binomial distributions, then general host–parasite models predict that the aggregation coefficient of the parasite distribution will increase with the average intensity of infections. We fit negative binomial distributions to the frequency distributions of ticks on hosts, and find that the estimated aggregation coefficient k increases with increasing average tick density. This pattern indirectly implies that one or more vital rates of the tick population must be changing with increasing tick density, because mortality rates of the tick's main host, the sleepy lizard, Tiliqua rugosa, are unaffected by changes in tick burdens. Our second piece of evidence is a re‐analysis of experimental data on the attachment success of individual ticks to lizard hosts using generalized linear modelling. The probability of successful engorgement decreases with increasing numbers of ticks attached to a host. This is direct evidence of a density‐dependent process that could lead to an increase in the aggregation coefficient of tick distributions described earlier. The population‐scale increase in the aggregation coefficient is indirect evidence of a density‐dependent process or processes sufficiently strong to produce a population‐wide pattern, and thus also likely to influence population regulation. The direct observation of a density‐dependent process is evidence of at least part of the responsible mechanism.     

Population dynamics of the chestnut gall-wasp,Dryocosmus kuriphilus Yasumatsu (Hymenoptera: Cynipidae)

Summary As a part of serial study on population dynamics of the chestnut gall-wasp,Dryocosmus kuriphilus, analyses of the distribution of eggs, gall-cells and emergent holes were made from the statistical point of view. Many of distributions of the eggs per bud could be described by the truncated Poisson, but some cases showed slight overdispersion than expected by chance. Because of no linear increase of with increasing (expected mean for complete sample), however, the truncated negative binomial seemed to be not so available for whole series. Distributions of the gall-cells and the emergent holes were, on the other hand, well described by the truncated Poisson distribution when the observed frequency was calculated for respective trees. Negative-binomial tendency found in distributions from some stations consisted of a certain number of trees would be resulted from mixture of Poisson populations with different means. Random or slightly concentrated oviposition and random mortality within galls was thus supposed for future study unless the proof favouring other distribution models would appear.     

Measures of spatial heterogeneity for species occurrence or disease incidence with finite-counts

We propose two types of indices with finite-count correction to measure the spatial heterogeneity of binary characteristics of organisms, such as occurrence or non-occurrence of organisms and infected or non-infected plants. We consider the following two examples: plant occurrence in a grassland community, and yellow dwarf disease infection in a rice field. For the grassland community, N quadrats comprising n cells of equal area, were set at random sites in a grassland, and the occurrence of a given species A in each of n cells was recorded. For disease infection, N quadrats, each consisting of n rice plants, were set at random sites in a paddy field, and the number of plants infected with yellow dwarf virus in each quadrat were counted. In these examples, since the number of cells in a quadrat is finite, neither occurrence nor incidence increase infinitely, unlike the number of aphids on a maize leaf. The first category of index belongs to the mean : variance ratio type. The estimated index value for occurrence (or incidence) is the same as that for non-occurrence (or non-incidence). The second category belongs to the k-type of a negative binomial distribution. If some random plants die or recover from the disease then the expected value of the second type of index does not change. For n, the current indices approach the mean : variance ratio and the inverse of k in a negative binomial distribution, respectively. This indicates that these indices are suitable for binary data sets.