Mixed Poisson Distribution Research Articles

POISSON-WEIGHTED EXPONENTIAL UNIVARIATE VERSION AND REGRESSION MODEL WITH APPLICATIONS

This study introduces a new two-parameter mixed Poisson distribution, namely Poisson-Weighted Exponential (P-WE), which is obtained by mixing Poisson distribution with a new class of weighted exponential distribution. The new P-WE distribution provides a more flexible alternative for modelling over dispersed count data compared to Poisson distr ibution. The estimation procedures of P-WE distribu tion via method of moments and maximum likelihood are provided. This study also introduces P-WE regression model which can be fitted to over dispersed count d ata with covariates. The P-WE distribution and P-WE regression model are fitted to two sets of count da ta.

Stochastic sir epidemic on a reed-frost random intersection graph

Let us consider a model for the spread of a stochastic SIR epidemic on a network of individuals described by a random intersection graph. The number of cliques a typical individual belongs to follows a mixed-Poisson distribution, as does the size of a typical clique. A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. A branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. Heuristically-motivated branching process approximations are described, which lead to a threshold parameter for the model and methods for calculating the probability of a major outbreak. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.

A stochastic model of cell survival for high-Z nanoparticle radiotherapy

The authors present a stochastic framework for the assessment of cell survival in gold nanoparticle radiotherapy. The authors derive the equations for the effective macroscopic dose enhancement for a population of cells with nonideal distribution of gold nanoparticles (GNP), allowing different number of GNP per cell and different distances with respect to the cellular target. They use the mixed Poisson distribution formalism to model the impact of the aforementioned physical factors on the effective dose enhancement. The authors show relatively large differences in the estimation of cell survival arising from using approximated formulae. They predict degeneration of the cell killing capacity due to different number of GNP per cell and different distances with respect to the cellular target. The presented stochastic framework can be used in interpretation of experimental cell survival or tumor control probability studies.

Specification Tests for the Poisson Distribution with a Stochastic Parameter

A new method is proposed for testing the mixed Poisson model which employs the well known identity between the probability generating function of the mixed Poisson distribution and the Laplace transform of the mixing distribution. The method utilizes a weighted L2-type statistic which may be easily calculated and the resulting test is consistent against general alternatives. A limit statistic is shown to be related to earlier methods which employ moment-- based contrasts. Some simulation results are also shown.

A class of bivariate negative binomial distributions with different index parameters in the marginals

In this paper, we consider a new class of bivariate negative binomial distributions having marginal distributions with different index parameters. This feature is useful in statistical modelling and simulation studies, where different marginal distributions and a specified correlation are required. This feature also makes it more flexible than the existing bivariate generalizations of the negative binomial distribution, which have a common index parameter in the marginal distributions. Various interesting properties, such as canonical expansions and quadrant dependence, are obtained. Potential application of the proposed class of bivariate negative binomial distributions, as a bivariate mixed Poisson distribution, and computer generation of samples are examined. Numerical examples as well as goodness-of-fit to simulated and real data are also given here in order to illustrate the application of this family of bivariate negative binomial distributions.

Dynamic Random Networks in Dynamic Populations

We consider a random network evolving in continuous time in which new nodes are born and old may die, and where undirected edges between nodes are created randomly and may also disappear. The node population is Markovian and so is the creation and deletion of edges, given the node population. Each node is equipped with a random social index and the intensity at which a node creates new edges is proportional to the social index, and the neighbour is either chosen uniformly or proportional to its social index in a modification of the model. We derive properties of the network as time and the node population tends to infinity. In particular, the degree-distribution is shown to be a mixed Poisson distribution which may exhibit a heavy tail (e.g. power-law) if the social index distribution has a heavy tail. The limiting results are verified by means of simulations, and the model is fitted to a network of sexual contacts.

Poisson approximation of the mixed Poisson distribution with infinitely divisible mixing law

In this work, explicit upper bounds are provided for the Kolmogorov and total variation distances between the mixed Poisson distribution with infinitely divisible mixing law and the Poisson distribution. If μ and σ 2 are the mean and variance of the mixing distribution respectively, then the bounds provided here are asymptotically equal to σ 2 / ( 2 μ 2 π e ) and σ 2 / ( μ 2 π e ) for the Kolmogorov and the total variation distance respectively when μ → ∞ and σ 2 is fixed. Finally, as an application, the Poisson approximation of the negative Binomial distribution is considered.

SOFTWARE RELIABILITY MODELING BASED ON MIXED POISSON DISTRIBUTIONS

This paper considers a novel modeling framework of software reliability models (SRMs). The proposed SRMs are based on the mixed Poisson distribution (MPD), which can involve the non-homogeneous Poisson process (NHPP) based SRMs completely, but are not always equivalent to them. More precisely, the MPD-based SRMs provide a mixture of NHPPs, and their statistical properties follows the mixed Poisson process. We develop a parameter estimation method for the MPD-based SRMs based on EM algorithm. In numerical examples, we mainly investigate the difference between conventional NHPP-based SRMs and MPD-based SRMs in the viewpoints of estimating parameters and software reliability.

Estimators of Tree Species Richness: An Assessment for Central and Eastern Canada

A total of 11 sample-based estimators of tree species richness (S) are evaluated in terms of accuracy and precision in a Monte Carlo simulated simple random sampling from 39,779 forest inventory plots with 7.8 million trees belonging to 85 species. The plots represent a 108 million hectare forested region in central and eastern Canada. Sample sizes varied from 50 to 800. A weighted index combining estimates of accuracy and precision identified Chao's first estimator (CHAO1) as overall best with an estimator based on the assumption of a gamma mixed Poisson distribution of species occurrence as a close runner-up. The observed sample species richness was almost always the most negatively biased estimate. A sample size of 400-700 conventional fixed area forest inventory plots are needed to produce results with bias <20%.

On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China

Heterogeneity in sexual behavior is known to play an important role in the spread of HIV. In 1986, a mathematical model based on ordinary differential equations was introduced to take into account the distribution of sexual activity. Assuming proportionate mixing, it was shown that the basic reproduction number R0 determining the epidemic threshold was proportional to M +V/M, where M is the mean and V the variance of the distribution. In the present paper, we notice that this theoretical distribution is different from the one obtained in behavioral surveys for the number of sexual partnerships over a period of length tau. The latter is a "mixed Poisson distribution" whose mean m and variance v are such that M = m/tau and V = (v-m)/tau(2). So M+V/M =(m+ v/m-1)/tau. This way, we improve the link between theory and data for sexual activity models of HIV/AIDS epidemics. As an example, we consider data concerning sex workers and their clients in Yunnan, China, and find an upper bound for the geometric mean of the transmission probabilities per partnership in this context.

Exponential families of mixed Poisson distributions

If I = ( I 1 , … , I d ) is a random variable on [ 0 , ∞ ) d with distribution μ ( d λ 1 , … , d λ d ) , the mixed Poisson distribution MP ( μ ) on N d is the distribution of ( N 1 ( I 1 ) , … , N d ( I d ) ) where N 1 , … , N d are ordinary independent Poisson processes which are also independent of I . The paper proves that if F is a natural exponential family on [ 0 , ∞ ) d then MP ( F ) is also a natural exponential family if and only if a generating probability of F is the distribution of v 0 + v 1 Y 1 + ⋯ + v q Y q for some q ⩽ d , for some vectors v 0 , … , v q of [ 0 , ∞ ) d with disjoint supports and for independent standard real gamma random variables Y 1 , … , Y q .

Compound Mixed Poisson Distributions II

The concept of compound mixed Poisson distributions in actuarial science is used to represent such variables as the total amount of claims or losses payable by an insurer. This paper provides the second part of the list of approximate forms for the compound mixed Poisson distribution (following up from Nadarajah and Kotz (2006)). The calculations involve use of several special functions and their properties. We believe that the results will serve as an important reference in actuarial science.

Compound mixed Poisson distributions I

The concept of compound mixed Poisson distributions in actuarial science is used to represent such variables as the total amount of claims or losses payable by an insurer. In this paper, comprehensive collections of approximate forms are derived for the compound mixed Poisson distribution. The calculations involve use of several special functions and their properties. We believe that the results will serve as an important reference in actuarial science.

Chromosomal biodosimetry by unfolding a mixed Poisson distribution: a generalized model

Purpose : The current chromosomal biodosimetry system is practically limited to the estimation of equivalent whole-body dose in the low or moderate dose range. To circumvent this problem, a new dosimetry system was developed which was effective over a wide dose range and dose gradient. Materials and methods : In order to establish a quantitative formalism of a comprehensive dose-response kinetics, chromosome aberration frequencies were studied in human peripheral blood lymphocytes irradiated in vitro with γ-rays in a dose range of 0.01-50Gy. Using the dose-yield relationship thus established, a new model of biodosimetry was developed which involved unfolding the chromosome aberration distribution into a mixed Poisson distribution and thence into a dose-distribution profile. The model was then tested with chromosome aberration data in lymphocytes irradiated in vitro, simulated partial body irradiation, accidental radiation exposure, therapeutic local irradiation and protracted exposure by internal deposit of Thorotrast. Results : The dose-yield relations over a wide dose range fit satisfactorily a multiparametric dose-response curve, which included the first approximation kinetics of linear-quadratic dose-response and its distortion by mutual interference among multiple breaks on a chromosome arm. The unfolding strategy using these dose-response kinetics was successfully applied to the various types of radiation exposures; the reconstructed dose profile showing a sharp unimodal peak for moderate or high doses of γ-rays in vitro irradiation, broadening for high-linear energy transfer radiation, discrimination of doses in the mixed culture of irradiated and unirradiated blood, and reasonable dose distribution for the in vivo exposures. Conclusion : A novel biodosimetry system was developed. The system is innovative in that it provides information not only on the macroscopic dose inhomogeneity, but also on the microscopic spectrum of doses stemming from the variability of energy transfer by charged particles as well as the multiple ionization events to which the target cell nucleus is exposed.

Chromosomal biodosimetry by unfolding a mixed Poisson distribution: a generalized model

Purpose : The current chromosomal biodosimetry system is practically limited to the estimation of equivalent whole-body dose in the low or moderate dose range. To circumvent this problem, a new dosimetry system was developed which was effective over a wide dose range and dose gradient. Materials and methods : In order to establish a quantitative formalism of a comprehensive dose-response kinetics, chromosome aberration frequencies were studied in human peripheral blood lymphocytes irradiated in vitro with γ-rays in a dose range of 0.01-50Gy. Using the dose-yield relationship thus established, a new model of biodosimetry was developed which involved unfolding the chromosome aberration distribution into a mixed Poisson distribution and thence into a dose-distribution profile. The model was then tested with chromosome aberration data in lymphocytes irradiated in vitro, simulated partial body irradiation, accidental radiation exposure, therapeutic local irradiation and protracted exposure by internal deposit of Thorotrast. Results : The dose-yield relations over a wide dose range fit satisfactorily a multiparametric dose-response curve, which included the first approximation kinetics of linear-quadratic dose-response and its distortion by mutual interference among multiple breaks on a chromosome arm. The unfolding strategy using these dose-response kinetics was successfully applied to the various types of radiation exposures; the reconstructed dose profile showing a sharp unimodal peak for moderate or high doses of γ-rays in vitro irradiation, broadening for high-linear energy transfer radiation, discrimination of doses in the mixed culture of irradiated and unirradiated blood, and reasonable dose distribution for the in vivo exposures. Conclusion : A novel biodosimetry system was developed. The system is innovative in that it provides information not only on the macroscopic dose inhomogeneity, but also on the microscopic spectrum of doses stemming from the variability of energy transfer by charged particles as well as the multiple ionization events to which the target cell nucleus is exposed.

A state space model for multivariate longitudinal count data

We propose a nonstationary state space model for multivariate longitudinal count data driven by a latent gamma Markov process. The Poisson counts are assumed to be conditionally independent given the latent process, both over time and across categories. We consider a regression model where time-varying covariates may enter via either the Poisson model or the latent gamma process. Estimation is based on the Kalman smoother, and we consider analysis of residuals from both the Poisson model and the latent process. A reanalysis of Zeger's (1988) polio data shows that the choice between a stationary and nonstationary model is crucial for the correct assessment of the evidence of a long-term decrease in the rate of U.S. polio infection. Keywords:EM algorithm; Estimation function; Generalised linear model; Kalman filter; Kalman smoother; Latent process; Mixed Poisson distribution; Overdispersion; Regression model; Residual analysis; Time-varying covariate.

Mixed Poisson distributions tail equivalent to their mixing distributions

A mixed Poisson distribution can have an upper tail asymptotically equal to the upper tail of its mixing distribution. Two broad classes of mixing distributions that generate mixed Poisson distributions with this property are identified: unbounded, non-negative distributions with mild regularity conditions that satisfy either (a) the von Mises condition for the Fréchet extreme value domain of attraction; or (b) the von Mises condition for the Gumbel extreme value domain of attraction and have hazard rates f( t)/(1 − F( t)) of order o( t − b ) for some δ ⩾ 1 2 as t → ∞. The Pareto distribution is prototypical of class (a) and the lognormal of class (b).

Word length in Portuguese texts*

The present paper tries to analyse the empirical word length distribution in Portuguese texts and asks whether it follows some probabilistic model. The hypothesis that word length distributions in texts are not chaotic, but abiding by specific laws is already proven for many different languages and therefore it forms the basis for an examination like the present one (cf., Grotjahn, 1982; Grotjahn & Altmann, 1993; Wimmer, Köhler, Grotjahn & Altmann, 1994; Wimmer, Best & Altmann, 1996). Because of different boundary conditions like the given language, individual style, functional style, etc., these laws lead to different theoretical models of word length distributions (cf., Grotjahn & Altmann; 1993, p. 142). That is the reason why only in very rare cases does the word length distribution in an individual language follow only one model. After the examination of word lengths in Brazilian‐Portuguese texts (cf., Ziegler, 1996), the present paper is until now the second one that deals with a variety of the Portuguese language. Therefore, in a further step, we will try to compare the results of the word length distribution in Brazilian‐Portuguese texts with those of this analysis to find out possible developments as well as common characteristic features and differences within one individual language that is subjected to different linguistic influences and social environments. We try to examine the European‐Portuguese texts by the model of the 1‐displaced mixed Poisson distribution, because nearly all of the Brazilian‐Portuguese texts followed this theoretical distribution as well.

A note on the mixed poisson formulation of the poisson-inverse gaussian distribution

The Poisson–inverse Gaussian distribution is expressed in terms of a mixed Poisson distribution with the inverted gamma as mixing distribution. A simple approximation by Taylor expansion is derived for the Poisson–inverse Gaussian probabilities from this mixing expression. Based on this expansion, the asymptotic behaviour is considered. Shaban's approximation (Communications in Statistics, A, 1981, pp. 1389–1399) of the Poisson-inverse Gaussian probability by its limiting case is examined, and it is shown that this approximation can be improved by using the inverted gamma mixing formulation.

Mixed Poisson approximation in the collective epidemic model

The collective epidemic model is a quite flexible model that describes the spread of an infectious disease of the Susceptible-Infected-Removed type in a closed population. A statistic of great interest is the final number of susceptibles who survive the disease. In the present paper, a necessary and sufficient condition is derived that guarantees the weak convergence of the law of this variable to a mixed Poisson distribution when the initial susceptible population tends to infinity, provided that the outbreak is severe in a certain sense. New ideas in the proof are the exploitation of a stochastic convex order relation and the use of a weak convergence theorem for products of i.i.d. random variables.