Poisson Binomial Distributions Research Articles

Aggregate Loss Models to Calculate Risk Measures

The concept of aggregate loss models pertains to a stochastic variable representing the total sum of all losses encountered within a set of insurance policies. In the non-life insurance sector, it is employed to assess the potential losses that an insurance company may face when claims made by policyholders exceed the allocated claim reserves. The purpose of studying aggregate loss models is to ascertain risk measures such as standard deviation of premium principles, value at risk (VaR), and conditional tail expectation (CTE). These steps aid insurance companies in the management and quantification of risks associated with aggregate losses. The standard deviation of premium principles is calculated analytically by substituting expected values and variances, while VaR is estimated using the Monte Carlo method to determine quantile values and confidence intervals. CTE is evaluated by computing the average losses that surpass the VaR threshold. These distributions and parameters require the Pareto distribution, which characterizes claim sizes, and the Poisson or Negative Binomial distribution, which factors in the number of claims. It is crucial to carefully consider the selection of the appropriate distribution, as it plays a significant role in determining the accuracy and reliability of the model. Furthermore, other influencing factors, such as loading factors and confidence intervals, should also be taken into account. These factors have the potential to significantly impact the quantification of risk arising from the model.

Epidemiology and Risk Factors of Eosinophilic Esophagitis in Japan: A Population-Based Study

Background and AimsEosinophilic esophagitis (EoE) has been increasingly diagnosed globally. However, there have been few general population-based studies in Asia. The aim of this study was to investigate EoE epidemiology in the Japanese general population. MethodsWe analyzed an employer-based health insurance claim database from January 2005 to September 2022. EoE cases were identified based on the International Classification of Diseases-tenth Revision code, K20.0. We calculated the incidence and prevalence of EoE using Poisson regression and binomial distribution, respectively. Using 10 matched controls for each EoE case, a nested case-control study was performed to identify potential risk factors for EoE. ResultsOf 15,200,895 individuals, 1,010 EoE cases were identified. The incidence and prevalence of EoE were 2.82 (95% CI 2.44–3.26) per 100,000 person-years and 10.68 (95% CI 10.01–11.37) per 100,000 people in 2022, nearly three and eight times as high as those in 2017, respectively. Smoking was associated with decreased risk of EoE (OR 0.45 (0.36 to 0.56), p<0.001) whereas alcohol consumption (OR 1.51 (1.21 to 1.88), p<0.001) were associated with increased risk of EoE along with several allergic conditions and psychiatric disorders. EoE was not related to either body mass index or lifestyle-related diseases such as hypertension, diabetes mellitus, hyperuricemia, and dyslipidemia. ConclusionThe incidence and prevalence of EoE in Japan have steadily increased over the past two decades. Nevertheless, EoE remains less common in Japan compared to the United States and Western Europe. Factors contributing to the epidemiology of EoE on a global basis may improve our understanding of the contribution of genetic and environmental risk factors.

Sequential sampling plan for the adults of Conotrachelus psidii (Marshall, 1922) (Coleoptera: Curculionidae) in guava crop

AbstractUnderstanding the spatial distribution and population fluctuations of insect pests facilitates the development of effective control strategies for a given crop. Conotrachelus psidii (Coleoptera: Curculionidae) is an important guava pest; however, no sampling plans are available for its management. Therefore, in this study, we aimed to (i) determine the spatial distribution pattern of C. psidii adults and (ii) develop a sequential sampling plan for pest control in guava crops. Evaluations began at 120 days after pruning and were performed weekly. To sample C. psidii adults, a white tarp was placed under the plants, covering the entire canopy. Plant branches were then shaken to dislodge the adult insects on the tarp, which were then identified and counted. Using the data obtained in each sampling, dispersion indices were determined and tests of adjustments to the theoretical Poisson and negative binomial frequency distributions were performed. The calculated dispersion indices revealed that the individuals in the studied populations exhibited aggregated dispersion. Notably, only four samples showed adjustment to the Poisson distribution, whereas 18 samples showed adjustment to the negative binomial distribution in the field, confirming the aggregated pattern of the population spatial arrangement. The sequential sampling plan developed for C. psidii revealed the maximum and minimum numbers of sample units expected for decision‐making as 45 and 15, respectively.

Spatial distribution of lower intertidal decapods on the northern Patagonian coast (Pelluhuín beach, Puerto Montt, 41°29′S, Chile)

Abstract The decapod fauna of Chile’s intertidal shores in inner seas south of 40°S has relatively low diversity because of the presence of low-salinity waters due to river inputs and glacial smelts; nevertheless it is possible that the same decapods species are found as on the northern and central Chilean coast. The aim of the present study was to determine the spatial distribution patterns of lower intertidal decapods on Pelluhuín beach, a small beach south of Puerto Montt, northern Patagonia. Data were obtained by counting individuals from random quadrants in intertidal zones; to the obtained data the variance/mean ratio was applied to determine if the specimens have a random, aggregate or uniform distribution, which are associated with Poisson, negative binomial or positive binomial distributions respectively. Among four of the species observed, a uniform distribution (positive binomial) was reported, and one had an aggregated pattern (negative binomial). The sites correspond to rocky shores in semi-urban zones, and in a protected zone. Our results on the interpretative probabilistic models of aggregated distribution patterns agree with previously reported observations of decapods on the rocky shores of Northern and Central Chile, specifically in interpretative probabilistic models.

Telematics combined actuarial neural networks for cross-sectional and longitudinal claim count data

AbstractWe present novel cross-sectional and longitudinal claim count models for vehicle insurance built upon the combinedd actuarial neural network (CANN) framework proposed by Wüthrich and Merz. The CANN approach combines a classical actuarial model, such as a generalized linear model, with a neural network. This blending of models results in a two-component model comprising a classical regression model and a neural network part. The CANN model leverages the strengths of both components, providing a solid foundation and interpretability from the classical model while harnessing the flexibility and capacity to capture intricate relationships and interactions offered by the neural network. In our proposed models, we use well-known log-linear claim count regression models for the classical regression part and a multilayer perceptron (MLP) for the neural network part. The MLP part is used to process telematics car driving data given as a vector characterizing the driving behavior of each insured driver. In addition to the Poisson and negative binomial distributions for cross-sectional data, we propose a procedure for training our CANN model with a multivariate negative binomial specification. By doing so, we introduce a longitudinal model that accounts for the dependence between contracts from the same insured. Our results reveal that the CANN models exhibit superior performance compared to log-linear models that rely on manually engineered telematics features.

Stein’s identities and the related topics: an instructive explanation on shrinkage, characterization, normal approximation and goodness-of-fit

The Stein-type identities are widely recognized for their substantial utility and potency in deriving shrinkage estimators improving on crude estimators in normal, gamma, Poisson, and negative binomial distributions. Additionally, these identities serve to characterize these distributions themselves. The Stein identities are also used to demonstrate normal approximation. Moreover, they are instrumental in constructing statistical tests to assess the goodness-of-fit for normality, exponentiality, and Poissonity of distributions. This article offers an instructive and comprehensive explanation of the applications of Stein-type identities in the aforementioned contexts.

A heavy-tailed model for analyzing miRNA-seq raw read counts.

This article addresses the limitations of existing statistical models in analyzing and interpreting highly skewed miRNA-seq raw read count data that can range from zero to millions. A heavy-tailed model using discrete stable distributions is proposed as a novel approach to better capture the heterogeneity and extreme values commonly observed in miRNA-seq data. Additionally, the parameters of the discrete stable distribution are proposed as an alternative target for differential expression analysis. An R package for computing and estimating the discrete stable distribution is provided. The proposed model is applied to miRNA-seq raw counts from the Norwegian Women and Cancer Study (NOWAC) and the Cancer Genome Atlas (TCGA) databases. The goodness-of-fit is compared with the popular Poisson and negative binomial distributions, and the discrete stable distributions are found to give a better fit for both datasets. In conclusion, the use of discrete stable distributions is shown to potentially lead to more accurate modeling of the underlying biological processes.

Investing in March Madness: An Examination of The Relationship Between Sports Betting and Portfolio Construction

This study focuses on three basic ideas: (1) sports betting should be considered an asset class; (2) sports betting contests, with two teams or contenders, are well described by a recently introduced probability distribution, the generalized Poisson binomial (GPB) distribution; and (3) Modern Portfolio Theory (MPT), Post-Modern Portfolio Theory (PMPT), and the Kelly criterion can be applied to yield optimal risk-return portfolios in such two contender environments. For the PMPT application, a unique quadratic-binary programming model is developed. March Madness data, based on the NCAA men’s annual championship basketball tournament, will provide examples of these portfolio theory approaches.

Modelling and diagnostic tests for Poisson and negative-binomial count time series

AbstractWhen modelling unbounded counts, their marginals are often assumed to follow either Poisson (Poi) or negative binomial (NB) distributions. To test such null hypotheses, we propose goodness-of-fit (GoF) tests based on statistics relying on certain moment properties. By contrast to most approaches proposed in the count-data literature so far, we do not restrict ourselves to specific low-order moments, but consider a flexible class of functions of generalized moments to construct model-diagnostic tests. These cover GoF-tests based on higher-order factorial moments, which are particularly suitable for the Poi- or NB-distribution where simple closed-form expressions for factorial moments of any order exist, but also GoF-tests relying on the respective Stein’s identity for the Poi- or NB-distribution. In the time-dependent case, under mild mixing conditions, we derive the asymptotic theory for GoF tests based on higher-order factorial moments for a wide family of stationary processes having Poi- or NB-marginals, respectively. This family also includes a type of NB-autoregressive model, where we provide clarification of some confusion caused in the literature. Additionally, for the case of independent and identically distributed counts, we prove asymptotic normality results for GoF-tests relying on a Stein identity, and we briefly discuss how its statistic might be used to define an omnibus GoF-test. The performance of the tests is investigated with simulations for both asymptotic and bootstrap implementations, also considering various alternative scenarios for power analyses. A data example of daily counts of downloads of a TeX editor is used to illustrate the application of the proposed GoF-tests.

The Applications of Generalized Poisson Regression Models to Insurance Claim Data

Predictive modeling has been widely used for insurance rate making. In this paper, we focus on insurance claim count data and address their common issues with more flexible modeling techniques. In particular, we study the zero-inflated and hurdle-generalized Poisson and negative binomial distributions in a functional form for modeling insurance claim count data. It is shown that these models are useful in addressing the problem of excess zeros and over-dispersion of the claim count variable. In addition, we show that including the exposure as a covariate in both the zero and the count part of the model is an effective approach to incorporating exposure information in zero-inflated and hurdle models. We illustrate the effectiveness and versatility of the introduced models using three real datasets. The results suggest their promising applications in insurance risk classification and beyond.

Bayesian CART models for insurance claims frequency

The accuracy and interpretability of a (non-life) insurance pricing model are essential qualities to ensure fair and transparent premiums for policy-holders, that reflect their risk. In recent years, classification and regression trees (CARTs) and their ensembles have gained popularity in the actuarial literature, since they offer good prediction performance and are relatively easy to interpret. In this paper, we introduce Bayesian CART models for insurance pricing, with a particular focus on claims frequency modelling. In addition to the common Poisson and negative binomial (NB) distributions used for claims frequency, we implement Bayesian CART for the zero-inflated Poisson (ZIP) distribution to address the difficulty arising from the imbalanced insurance claims data. To this end, we introduce a general MCMC algorithm using data augmentation methods for posterior tree exploration. We also introduce the deviance information criterion (DIC) for tree model selection. The proposed models are able to identify trees which can better classify the policy-holders into risk groups. Simulations and real insurance data will be used to illustrate the applicability of these models.

Beta-Geometric Regression for Modeling Count Data on First Antenatal Care Visit (ANC) with Application

Introduction: Little attention has been paid to modeling count data with the geometric distribution. There are many real-life phenomena with a constant probability of first success. However, in practice, the probability of the first success may vary, making simple geometric models unsuitable for modeling such data. One can assume one of many continuous distributions for modeling the probability of first success with the parameter space [0, 1]. In this respect Beta distribution defined on the standard unit interval [0,1] is the most useful distribution due to its ability to accommodate a wide range of shapes. Thus, in this paper, by mixing Beta and geometric distribution, we developed a Beta-geometric distribution for modeling the count data through application to real-life count data on time to the first antenatal care (ANC) visit.&#x0D; Methods: The estimation of the distribution parameters using the method of moments, maximum likelihood estimation (MLE) method, and Bayesian estimation approach are provided. Based on the Beta-geometric distribution, we developed a new Beta-geometric regression model for analyzing count data that follow the geometric distribution. The goodness of fit of the derived model has been tested using real data on time to the first ANC visit.&#x0D; Results: Beta-geometric distribution has a simple form for its probability mass function (pmf), and is flexible in capturing both underdispersion and overdispersion that may present in count data. It was found that the proposed Beta-geometric regression model fit the count data on the first ANC visit better than simple geometric distribution or Negative Binomial distribution.&#x0D; Conclusion: Unlike the Poisson or Negative Binomial distribution, Beta-geometric distribution does not need an additional parameter to accommodate underdispersion or overdispersion and thus could be a flexible choice for analyzing any count data. The goodness of fit test of the Beta-geometric model provides better fitting of the model to real data on time to first ANC visit than geometric or Negative binomial models.

Plant growth forms influence sandhill longleaf pine regeneration

Longleaf pine researchers have quantified many aspects of regeneration, growth and yield, but have focused on productive sites such as pine flatwoods and upland pine. Sandhill longleaf pine woodlands represent >40 % of remnant stands of longleaf pine ecosystems; most have no history of agriculture due to poor site productivity, yet many have been degraded by resin collection, grazing and fire exclusion. Restoration of structure and function of sandhill longleaf pine depends on understanding early stand-development processes, which are governed by fire, propagule availability, and competition from a range of overstory tree species not limited to longleaf pine. We analyzed a 16-year monitoring dataset from Eglin Air Force Base in northwest Florida, USA, focusing on how four tree growth forms influenced longleaf pine juvenile stages of seedling and sapling numbers and ingrowth to the subadult size class. Analyses were done with generalized linear mixed models using Poisson, normal, or negative binomial error distributions according to the response variable.Restoration activities were effective at reducing numbers of sapling evergreen hardwoods, and basal area of overstory sand pine and other non-longleaf conifers was reduced by 35 % over the 16-year study. We observed threshold values for adult longleaf pine effects on seedlings and ingrowth; seedling numbers were maximized at 7.8 m2/ha, and ingrowth was maximum at 5.1 m2/ha. Seedling density was strongly governed by mast year. Sand pine, an invasive native pine, was detrimental to longleaf pine at all juvenile life stages even at low basal area. Deciduous oaks, which have been observed to facilitate longleaf pine seedling establishment, instead facilitated ingrowth to the subadult size class. Evergreen hardwoods, which may dampen fire behavior, only decreased longleaf pine density at the sapling stage. Ongoing silvicultural restoration activities should continue to focus on frequent fire and sand pine removal; deciduous oaks should be retained for their facilitative value; and removal of evergreen hardwoods should not be a high priority as long as prescribed fire can propagate in these lower productivity sites.

Process Capability and Performance Indices for Discrete Data

Process capability and performance indices (PCIs and PPIs) are used in industry to provide numerical measures for the capability and performance of several processes. The majority of the literature refers to PCIs and PPIs for continuous data. The aim of this paper is to compute the classical indices for discrete data following Poisson, binomial or negative binomial distribution using various transformation techniques. A simulation study under different situations of a process and comparisons with other existing PCIs for discrete data are also presented. The methodology of computing the indices is easy to use, and as a result, one can have an assessment of the process capability and performance without difficulty. Three examples are further provided to illustrate the application of the transformation techniques.

Necessary and sufficient conditions for exact closures of epidemic equations on configuration model networks

We prove that it is possible to obtain the exact closure of SIR pairwise epidemic equations on a configuration model network if and only if the degree distribution follows a Poisson, binomial, or negative binomial distribution. The proof relies on establishing the equivalence, for these specific degree distributions, between the closed pairwise model and a dynamical survival analysis (DSA) model that was previously shown to be exact. Specifically, we demonstrate that the DSA model is equivalent to the well-known edge-based Volz model. Using this result, we also provide reductions of the closed pairwise and Volz models to a single equation that involves only susceptibles. This equation has a useful statistical interpretation in terms of times to infection. We provide some numerical examples to illustrate our results.

A zero-inflated endemic-epidemic model with an application to measles time series in Germany.

Count data with an excess of zeros are often encountered when modeling infectious disease occurrence. The degree of zero inflation can vary over time due to nonepidemic periods as well as by age group or region. A well-established approach to analyze multivariate incidence time series is the endemic-epidemic modeling framework, also known as the HHH approach. However, it assumes Poisson or negative binomial distributions and is thus not tailored to surveillance data with excess zeros. Here, we propose a multivariate zero-inflated endemic-epidemic model with random effects that extends HHH. Parameters of both the zero-inflation probability and the HHH part of this mixture model can be estimated jointly and efficiently via (penalized) maximum likelihood inference using analytical derivatives. We found proper convergence and good coverage of confidence intervals in simulation studies. An application to measles counts in the 16 German states, 2005-2018, showed that zero inflation is more pronounced in the Eastern states characterized by a higher vaccination coverage. Probabilistic forecasts of measles cases improved when accounting for zero inflation. We anticipate zero-inflated HHH models to be a useful extension also for other applications and provide an implementation in anRpackage.

Probabilistic tensor decomposition extracts better latent embeddings from single-cell multiomic data.

Single-cell sequencing technology enables the simultaneous capture of multiomic data from multiple cells. The captured data can be represented by tensors, i.e.the higher-rank matrices. However, the existing analysis tools often take the data as a collection of two-order matrices, renouncing the correspondences among the features. Consequently, we propose a probabilistic tensor decomposition framework, SCOIT, to extract embeddings from single-cell multiomic data. SCOIT incorporates various distributions, including Gaussian, Poisson, and negative binomial distributions, to deal with sparse, noisy, and heterogeneous single-cell data. Our framework can decompose a multiomic tensor into a cell embedding matrix, a gene embedding matrix, and an omic embedding matrix, allowing for various downstream analyses. We applied SCOIT to eight single-cell multiomic datasets from different sequencing protocols. With cell embeddings, SCOIT achieves superior performance for cell clustering compared to nine state-of-the-art tools under various metrics, demonstrating its ability to dissect cellular heterogeneity. With the gene embeddings, SCOIT enables cross-omics gene expression analysis and integrative gene regulatory network study. Furthermore, the embeddings allow cross-omics imputation simultaneously, outperforming current imputation methods with the Pearson correlation coefficient increased by 3.38-39.26%; moreover, SCOIT accommodates the scenario that subsets of the cells are with merely one omic profile available.

Local limit theorems for collective risk models

A probability model known as the collective risk model is used to describe the total claim from a portfolio of insurance contracts. It is essential to non-life insurance. Let N represent the number of claims and X1,X2,… represent the amount of loss in each claim where Xj′s are independent and identically distributed. For the collective risk model, the total claim is given by SN=X1+X2+⋯+XN. Local limit theorems estimate the probability at a particular point P(SN=k). In this paper, we provide local limit theorems for SN, where N is a random variable with a binomial, Poisson and negative binomial distribution. Our results give a better rate of convergence than Berry–Esseen’s Theorem. Explicit constants of the error bounds are also given.

Latent Growth Models for Count Outcomes: Specification, Evaluation, and Interpretation

The latent growth model (LGM) is a popular tool in the social and behavioral sciences to study development processes of continuous and discrete outcome variables. A special case are frequency measurements of behaviors or events, such as doctor visits per month or crimes committed per year. Probability distributions for such outcomes include the Poisson or negative binomial distribution and their zero-inflated extensions to account for excess zero counts. This article demonstrates how to specify, evaluate, and interpret LGMs for count outcomes using the Mplus program in the structural equation modeling framework. The foundations of LGMs for count outcomes are discussed and illustrated using empirical count data on self-reported criminal offenses of adolescents (N = 1,664; age 15–18). Annotated syntax and output are presented for all model variants. A negative binomial LGM is shown to best fit the crime growth process, outperforming Poisson, zero-inflated, and hurdle LGMs.

Poisson approximation in χ2 distance by the Stein-Chen approach

The main purpose of the paper is to investigate the possibility of applying the Stein-Chen approach to estimate the χ2 distance between Poisson distribution and Poisson binomial distribution. Earlier results concerning χ2 distance between the above mentioned distributions either used analytical approach heavily based on the analysis of the generating functions or on rather lengthy and complicated elementary calculations. Applying the Stein-Chen approach we succeed in providing a very quick proof of upper bounds for χ2 distance that are of comparable strength to the earlier estimates obtained by other approaches.