Mixed Poisson Distributions Research Articles

Phase transitions of composition schemes: Mittag-Leffler and mixed Poisson distributions

Phase transitions of composition schemes: Mittag-Leffler and mixed Poisson distributions

Comparison of Generalized Linear Model between Gamma and Tweedie Compound Response for Rainfall Prediction in Lampung Province

Rainfall plays a pivotal role in influencing agricultural production in Lampung province. The precision of rainfall predictions holds significant importance for enhancing agricultural yields in the region. One effective approach for modeling rainfall is Statistical Downscaling (SD), which employs statistical models to examine the correlation between large-scale (global) climatological data and small-scale (local) data. SD addresses the limitation of global scale data, such as the General Circulation Model (GCM), which lacks the resolution to directly forecast localized climate conditions like rainfall. Rainfall can be broadly categorized into continuous and discrete components. The continuous component delineates the intensity of rainfall, while the discrete component describes the occurrence of rain. Both components are integral to accurate rainfall predictions. The mixed Tweedie distribution, combining Gamma and Poisson distributions, is proficient in handling both continuous and discrete components of rainfall. GCMs commonly encounter multicollinearity issues in SD modeling, which can be mitigated through Principal Component Analysis. This study seeks to compare two regression models: the generalized linear model with a gamma response and the Tweedie compound response. Rainfall data from three distinct regions in Lampung province, representing high, medium, and lowlands, is utilized. The research findings indicate that, for high and lowlands, the Tweedie compound exhibits a smaller Root Mean Square Error of Prediction (RMSEP) compared to gamma. Conversely, in medium lands, gamma-GLM demonstrates a smaller RMSEP value than the Tweedie compound. Thus, the distribution of the Tweedie compound is better suited for use than Gamma-GLM, especially for high and lowland areas.

Recent Developments in Mixed Poisson Distributions

Mixed Poisson distributions are a class of distributions arising from the Poisson mean fluctuating as a random variable. Mixed Poisson distributions have been applied in diverse disciplines for modelling non-homogeneity in populations. This paper brings together recent work on this class of distributions with focus on specific models, computation and simulation, applications to stochastic and data modelling.

A new two-parameter discrete poisson-generalized Lindley distribution with properties and applications to healthcare data sets

Mixed-Poisson distributions have been used in many fields for modeling the over-dispersed count data sets. To open a new opportunity in modeling the over-dispersed count data sets, we introduce a new mixed-Poisson distribution using the generalized Lindley distribution as a mixing distribution. The moment and probability generating functions, factorial moments as well as skewness, and kurtosis measures are derived. Using the mean-parametrized version of the proposed distribution, we introduce a new count regression model which is an appropriate model for over-dispersed counts. The healthcare data sets are analyzed employing a new count regression model. We conclude that the new regression model works well in the case of over-dispersion.

Convergence and inference for mixed Poisson random sums

We study the limit distribution of partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixture between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between $$\alpha $$ -stable distributions and NEF laws is established. We propose the estimation of the NEF model parameters through the method of moments and also by the maximum likelihood method via an Expectation–Maximization algorithm. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators, and an empirical illustration of the financial market is presented.

Importance of Distributional Forms for the Assessment of Protozoan Pathogens Concentrations in Drinking-Water Sources.

The identification of appropriately conservative statistical distributions is needed to predict microbial peak events in drinking water sources explicitly. In this study, Poisson and mixed Poisson distributions with different upper tail behaviors were used for modeling source water Cryptosporidium and Giardia data from 30 drinking water treatment plants. Small differences (<0.5-log) were found between the "best" estimates of the mean Cryptosporidium and Giardia concentrations with the Poisson-gamma and Poisson-log-normal models. However, the upper bound of the 95% credibility interval on the mean Cryptosporidium concentrations of the Poisson-log-normal model was considerably higher (>0.5-log) than that of the Poisson-gamma model at four sites. The improper choice of a model may, therefore, mislead the assessment of treatment requirements and health risks associated with the water supply. Discrimination between models using the marginal deviance information criterion (mDIC) was unachievable because differences in upper tail behaviors were not well characterized with available data sets ( ). Therefore, the gamma and the log-normal distributions fit the data equally well but may predict different risk estimates when they are used as an input distribution in an exposure assessment. The collection of event-based monitoring data and the modeling of larger routine monitoring data sets are recommended to identify appropriately conservative distributions to predict microbial peak events.

Impact of Hydrometeorological Events for the Selection of Parametric Models for Protozoan Pathogens in Drinking-Water Sources.

Temporal variations in concentrations of pathogenic microorganisms in surface waters are well known to be influenced by hydrometeorological events. Reasonable methods for accounting for microbial peaks in the quantification of drinking water treatment requirements need to be addressed. Here, we applied a novel method for data collection and model validation to explicitly account for weather events (rainfall, snowmelt) when concentrations of pathogens are estimated in source water. Online in situ β-d-glucuronidase activity measurements were used to trigger sequential grab sampling of source water to quantify Cryptosporidium and Giardia concentrations during rainfall and snowmelt events at an urban and an agricultural drinking water treatment plant in Quebec, Canada. We then evaluate if mixed Poisson distributions fitted to monthly sampling data ( = 30 samples) could accurately predict daily mean concentrations during these events. We found that using the gamma distribution underestimated high Cryptosporidium and Giardia concentrations measured with routine or event-based monitoring. However, the log-normal distribution accurately predicted these high concentrations. The selection of a log-normal distribution in preference to a gamma distribution increased the annual mean concentration by less than 0.1-log but increased the upper bound of the 95% credibility interval on the annual mean by about 0.5-log. Therefore, considering parametric uncertainty in an exposure assessment is essential to account for microbial peaks in risk assessment.

Bounds for convergence rate in laws of large numbers for mixed Poisson random sums

In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev ζ-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous Rényi theorem) to a considerably wider class of random indices with mixed Poisson distributions including, e.g., those with the (generalized) negative binomial distribution.

A nonparametric method of estimation of the population size in capture-recapture experiments.

A recent method for estimating a lower bound of the population size in capture-recapture samples is studied. Specifically, some asymptotic properties, such as strong consistency and asymptotic normality, are provided. The introduced estimator is based on the empirical probability generating function (pgf) of the observed data, and it is consistent for count distributions having a log-convex pgf ( -class). This is a large family that includes mixed and compound Poisson distributions, and their independent sums and finite mixtures as well. The finite-sample performance of the lower bound estimator is assessed via simulation showing a better behavior than some close competitors. Several examples of application are also analyzed anddiscussed.

Flexible regression models for counts with high-inflation of zeros

In this paper, we introduce a flexible class of regression models for counts with high-inflation of zeros that cannot be predicted by the Poisson, the zero-inflated Poisson, the negative binomial and the Poisson-inverse Gaussian regression models. Our proposed flexible regression models are based on a class of zero-inflated mixed Poisson distributions and contain the zero-inflated negative binomial (ZINB) and the zero-inflated Poisson-inverse Gaussian (ZIPIG) distributions, as particular cases, among others. We consider regression structures for the mean, the dispersion, and the zero-inflation parameters. Consequently, we generalize existing models, such as the ZINB regression (with non-varying dispersion), and also open the possibility of introducing new models, such as the ZIPIG and the zero-inflated generalized hyperbolic secant regressions. We propose an Expectation-Maximization (in short EM) algorithm for estimating the parameters and the associated information matrix. Simulation results are presented to compare the finite-sample performance of our proposed EM-algorithm with a direct maximization of the log-likelihood function based on the GAMLSS approach. These simulated results show some advantages of our EM-algorithm concerning the GAMLSS proposal. We also discuss a measure of influence based on the EM approach and propose simulated envelopes for checking the adequacy of our zero-inflated regression models. An empirical application, about the number of roots produced by 270 micropropagated shoots of the columnar apple cultivar Trajan, illustrates the usefulness of the proposed class of regression models for dealing with count data presenting high-inflation of zeros and shows that one cannot use the GAMLSS approach in some practical situations due to numerical problems.

Convergence rates for the degree distribution in a dynamic network model

In the stochastic network model of Britton and Lindholm, the number of individuals evolves according to a supercritical linear birth and death process, and a random social index is assigned to each individual at birth, which controls the rate at which connections to other individuals are created. We derive a rate for the convergence of the degree distribution in this model towards the mixed Poisson distribution determined by Britton and Lindholm based on heuristic arguments. To do so, we deduce the degree distribution at finite time and derive an approximation result for mixed Poisson distributions to compute an upper bound for the total variation distance to the asymptotic degree distribution.

A general method of computing mixed Poisson probabilities by Monte Carlo sampling

Mixed Poisson distributions form an important class of distributions in applications. However, the application of many of these mixed Poisson distributions is hampered by the complicated probability distributions. The paper examines Monte Carlo sampling as a general technique for computation of mixed Poisson probabilities which is applicable to any mixed Poisson distribution with arbitrary mixing distribution. The accuracy and computational speed of this method is illustrated with the Poisson–inverse Gaussian distribution. The proposed method is then applied to compute probabilities of the Poisson–lognormal distribution, a popular species abundance model. It is also shown that in the maximum likelihood estimation of Poisson–lognormal parameters by E–M algorithm, the application of the proposed Monte Carlo computation in the algorithm avoids numerical problems.

Generalized negative binomial distributions as mixed geometric laws and related limit theorems*

In this paper, we study a wide and flexible family of discrete distributions, the so-called generalized negative binomial (GNB) distributions, which are mixed Poisson distributions with the mixing laws belonging to the class of generalized gamma (GG) distributions. This family was introduced by E.W. Stacy as a particular family of lifetime distributions containing gamma, exponential power, and Weibull distributions. These distributions seem to be very promising in the statistical description of many real phenomena and very convenient and almost universal models for the description of statistical regularities in discrete data. We study analytic properties of GNB distributions. We prove that a GG distribution is a mixed exponential distribution if and only if the shape and exponent power parameters are no greater than one. We write the mixing distribution explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative half-line. As a corollary, we obtain a representation for the GNB distribution as a mixed geometric distribution. We consider the corresponding scheme of Bernoulli trials with random probability of success. Within this scheme, we prove a random analog of the Poisson theorem establishing the convergence of mixed binomial distributions to mixed Poisson laws. We prove limit theorems for random sums of independent random variables in which the number of summands has the GNB distribution and the summands have both light- and heavy-tailed distributions. We obtain that the class of limit laws is wide enough and includes the so-called generalized variance gamma distributions. We obtain various representations for the limit laws in terms of mixtures of Mittag-Leffler, Linnik, or Laplace distributions. We prove limit theorems establishing the convergence of the distributions of statistics constructed from samples with random sizes obeying the GNB distribution to generalized variance gamma distributions. We also discuss some applications of GNB distributions in meteorology.

Flexible and Robust Mixed Poisson INGARCH Models

In this article, we propose a general class of INteger‐valued Generalized AutoRegressive Conditional Heteroskedastic (INGARCH) models based on a flexible family of mixed Poisson (MP) distributions. Our proposed class of count time series models contains the negative binomial (NB) INGARCH process as particular case and open the possibility to introduce new models such as the Poisson‐inverse Gaussian (PIG) and Poisson generalized hyperbolic secant processes. In particular, the PIG INGARCH model is an interesting and robust alternative to the NB model. We explore first‐order and second‐order stationary properties of our MPINGARCH models and provide expressions for the autocorrelation function and mean and variance marginals. Conditions to ensure strict stationarity and ergodicity properties for our class of INGARCH models are established. We propose an Expectation‐Maximization algorithm to estimate the parameters and obtain the associated information matrix. Further, we discuss two additional estimation methods. Monte Carlo simulation studies are considered to evaluate the finite‐sample performance of the proposed estimators. We illustrate the flexibility and robustness of the MPINGARCH models through two real‐data applications about number of cases of Escherichia coli and Campylobacter infections. This article contains a Supporting Information.

Bayesian Estimation of Earth’s Undiscovered Mineralogical Diversity Using Noninformative Priors

Recently, statistical distributions have been explored to provide estimates of the mineralogical diversity of Earth, and Earth-like planets. In this paper, a Bayesian approach is introduced to estimate Earth’s undiscovered mineralogical diversity. Samples are generated from a posterior distribution of the model parameters using Markov chain Monte Carlo simulations such that estimates and inference are directly obtained. It was previously shown that the mineral species frequency distribution conforms to a generalized inverse Gauss–Poisson (GIGP) large number of rare events model. Even though the model fit was good, the population size estimate obtained by using this model was found to be unreasonably low by mineralogists. In this paper, several zero-truncated, mixed Poisson distributions are fitted and compared, where the Poisson-lognormal distribution is found to provide the best fit. Subsequently, the population size estimates obtained by Bayesian methods are compared to the empirical Bayes estimates. Species accumulation curves are constructed and employed to estimate the population size as a function of sampling size. Finally, the relative abundances, and hence the occurrence probabilities of species in a random sample, are calculated numerically for all mineral species in Earth’s crust using the Poisson-lognormal distribution. These calculations are connected and compared to the calculations obtained in a previous paper using the GIGP model for which mineralogical criteria of an Earth-like planet were given.

Scale Mixtures of Frechet Distributions as Asymptotic Approximations of Extreme Precipitation

Based on the negative binomial model for the duration of wet periods measured in days, an asymptotic approximation is proposed for the distribution of the maximum daily precipitation volume within a wet period. This approximation has the form of a scale mixture of the Frechet distribution with the gamma mixing distribution and coincides with the distribution of a positive power of a random variable having the Snedecor-Fisher distribution. The proof of this result is based on the representation of the negative binomial distribution as a mixed geometric (and hence, mixed Poisson) distribution and limit theorems for extreme order statistics in samples with random sizes having mixed Poisson distributions. Some analytic properties of the obtained limit distribution are described. In particular, it is demonstrated that under certain conditions the limit distribution is mixed exponential and hence, is infinitely divisible. It is shown that under the same conditions the limit distribution can be represented as a scale mixture of stable or Weibull or Pareto or folded normal laws. The corresponding product representations for the limit random variable can be used for its computer simulation. Several methods are proposed for the estimation of the parameters of the distribution of the maximum daily precipitation volume. The results of fitting this distribution to real data are presented illustrating high adequacy of the proposed model. The obtained mixture representations for the limit laws and the corresponding asymptotic approximations provide better insight into the nature of mixed probability ("Bayesian") models.

Izračunavanje premije u osiguranju motornih vozila pomoću složenih Paoasonovih distribucija

Automobile insurance, particularly motor third liability insurance covers dominant share in the portfolio of non-life insurance in almost all countries of the world. The aim of this paper is to analyze the process of determining premium rates in automobile liability insurance using the bonus-malus system as one of the key instruments to increase traffic safety. The paper presents a brief analysis of the automobile liability insurance market in Serbia, with special reference to the bonus-malus system introduced in practice. Using a sample from the year 2015 provided by one of the biggest Serbian automobile liability insurer it has been proved that the mixed Poisson distributions are appropriate choice in the process of modeling of the afore mentioned tariff system. The main result of the study is the construction of bonus-malus system based on mixed Poisson models, where expected value principle is used to determine the net premium. According to the analysis, it has been concluded that during the creation of a bonus-malus system in practice it is important to use certain mathematical and statistical models, as well as that the existing tariff system of automobile liability insurance in Serbia can be improved by introducing additional assumptions.

Mixed Poisson INAR(1) processes

Overdispersion is a phenomenon commonly observed in count time series. Since Poisson distribution is equidispersed, the INteger-valued AutoRegressive (INAR) process with Poisson marginals is not adequate for modelling overdispersed counts. To overcome this problem, in this paper we propose a general class of first-order INAR processes for modelling overdispersed count time series. The proposed INAR(1) processes have marginals belonging to a class of mixed Poisson distributions, which are overdispersed. With this, our class of overdispersed count models have the known negative binomial INAR(1) process as particular case and open the possibility of introducing new INAR(1) processes, such as the Poisson-inverse Gaussian INAR(1) model, which is discussed here with some details. We establish a condition to our class of overdispersed INAR processes is well-defined and study some statistical properties. We propose estimators for the parameters and establish their consistency and asymptotic normality. A small Monte Carlo simulation to evaluate the finite-sample performance of the proposed estimators is presented and one application to a real data set illustrates the usefulness of our proposed overdispersed count processes.

On moment sequences and mixed Poisson distributions

In this article we survey properties of mixed Poisson distributions and probabilistic aspects of the Stirling transform: given a non-negative random variable (X) with moment sequence ((mu_s)_{sinmathbb{N}}) we determine a discrete random variable (Y), whose moment sequence is given by the Stirling transform of the sequence ((mu_s)_{sinmathbb{N}}), and identify the distribution as a mixed Poisson distribution. We discuss properties of this family of distributions and present a new simple limit theorem based on expansions of factorial moments instead of power moments. Moreover, we present several examples of mixed Poisson distributions in the analysis of random discrete structures, unifying and extending earlier results. We also add several entirely new results: we analyse triangular urn models, where the initial configuration or the dimension of the urn is not fixed, but may depend on the discrete time (n). We discuss the branching structure of plane recursive trees and its relation to table sizes in the Chinese restaurant process. Furthermore, we discuss root isolation procedures in Cayley trees, a parameter in parking functions, zero contacts in lattice paths consisting of bridges, and a parameter related to cyclic points and trees in graphs of random mappings, all leading to mixed Poisson-Rayleigh distributions. Finally, we indicate how mixed Poisson distributions naturally arise in the critical composition scheme of Analytic Combinatorics. <script type=text/javascript src=//cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML>

Using the proportion of barren samples as a proxy for minimum grade in a diamondiferous linear beach deposit - an application of the Nachman model

�7.,!*-* Over the past 80 years, the Namibian diamondiferous marine placer has been studied extensively to develop solutions for mining and sampling challenges. The types of studies include the statistical modelling of the distributions of the stone counts per sample; investigating the relationship between geology and the grade distribution; assessing the quality potential of the entrapment of the available diamond pulse; using predetermined acceptability of barren samples (zero proportion (Z p) samples) to model distributions; optimal sample sizes; and more. During early-stage project evaluation it is more important to find out if a particular area is likely to be above a specific cut-off grade than to focus on sampling for the purpose of accurate resource estimation. Previous work using mixed Poisson and Sichel distributions to model the abundant onshore diamond data has been very successful in modelling the long- tailed nature of these linear beach deposits. The means of these distrib- utions are, however, sensitive to extreme values. Technical and cost constraints prevent a similar scale of sample collection in an adjacent, geologically equivalent, submerged beach environment. A method not sensitive to extreme values is thus required to make early-stage assessments of the likelihood that the grade of a particular target is above a minimum cut-off grade. The Nachman model describes the functional relationship between the mean population density and proportion of barren patches (Z