An Adaptive Quadratic Transmuted Exponential Distribution: Unbounded Weight Parameter and Mixed Poisson Model Integrations

Generalized Linear Models (GLMs) are a modeling approach that allows the modeling of nonlinear behaviors and non-Gaussian distributions of residues. This approach is very useful for general insurance analysis, where the frequency of claims and the amount of claims distributions are usually non-Gaussian. In this article, the application of Poisson and Negative Binomial models to estimate the frequency of claims of auto insurance is discussed. The accuracy of the models was compared to choose the best model for determining pure insurance premiums using R software. The data used are a secondary dataset which is the motor vehicle insurance dataset from Sweden named dataOhlsson and the motor vehicle dataset from Australia named ausprivauto0405. The results of the exploration of the GLMs model are that Poisson's GLM and Negative Binomial models both are suitable models for estimating the number of claims for the dataOhlsson dataset. Both models have relatively similar parameter estimates, as well as the AIC and BIC values for the dataOhlsson dataset, however, both models are not suitable for estimating the number of claims for the ausprivauto0405 dataset. More investigation using different models is needed to ensure which model is more appropriate for estimating the frequency of insurance claims.

Predicting tree recruitment with negative binomial mixture models

Within non-life insurance pricing, an accurate evaluation of claim frequency, also known in theory as count data, represents an essential part in determining an insurance premium according to the policyholder’s degree of risk. Count regression analysis allows the identification of the risk factors and the prediction of the expected frequency of claims given the characteristics of policyholders. The aim of this paper is to verify several hypothesis related to the methodology of count data models and also to the risk factors used to explain the frequency of claims. In addition to the standard Poisson regression, Negative Binomial models are applied to a French auto insurance portfolio. The best model was chosen by means of the log-likelihood ratio and the information criteria. Based on this model, the profile of the policyholders with the highest degree of risk is determined

Count data models are developed for breeding applications to account for more variability than in a Poisson mixed effects model. A gamma distribution is assigned to Poisson parameters, thereby leading to a negative binomial model. The natural log of the expected value of the Poisson parameter is expressed as a linear function of fixed and random polygenic effects. The negative binomial and Poisson mixed models were compared in two simulations. In the first, marginal maximum likelihood (MML) estimates of genetic variances obtained under a Poisson sire model (PSM) and under a Poisson animal model (PAM), accounting for half-sib relationships, were different, contrary to what occurs in a Gaussian mixed linear model. MML estimates of genetic variance under a negative binomial sire model were less biased than estimates under a PSM, and had a slightly smaller mean squared error (MSE). The second simulation compared animal models in which the variance of the residuals was larger than the genetic variance. Empirical relative bias and MSE of MML estimates of genetic variance were larger under a PAM that ignored the residuals than under a negative binomial model. Differences in performance widened as genetic variance increased. An application to the analysis of number of artificial inseminations until conception in dairy heifers is presented to illustrate potential differences in genetic variance estimates under the two models.

Can Generalized Poisson model replace any other count data models? An evaluation

Count data models have found a wide variety of applications not only in applied economics and finance but also in diverse fields ranging from biometrics to political science. Poisson and negative binomial (NB) models have been extensively used in count data analysis. Two particular NB model specifications, NBI and NBII, have been especially popular. However, these models impose arbitrary restrictions on the relation between the conditional mean and variance of the dependent variable, limiting their generality. This study proposes tests for selection among the Poisson and NB models by formally demonstrating that the log likelihood function (LLF) of a general NB model parametrically nests the LLF of the Poisson, NBI and NBII as testable special cases. It also proposes estimation of the general NB model since it allows greater flexibility in the relationship between the mean and variance of the dependent variable than NBI and NBII. The empirical application, which uses micro‐level data on recreational boating, provides support for the paper's main theme. Tests clearly reject not only the Poisson, but also NBI and NBII, in favour of a different NB model, underscoring the importance of the general model specification.

ABSTRACTThe purpose of this paper is to develop a new linear regression model for count data, namely generalized-Poisson Lindley (GPL) linear model. The GPL linear model is performed by applying generalized linear model to GPL distribution. The model parameters are estimated by the maximum likelihood estimation. We utilize the GPL linear model to fit two real data sets and compare it with the Poisson, negative binomial (NB) and Poisson-weighted exponential (P-WE) models for count data. It is found that the GPL linear model can fit over-dispersed count data, and it shows the highest log-likelihood, the smallest AIC and BIC values. As a consequence, the linear regression model from the GPL distribution is a valuable alternative model to the Poisson, NB, and P-WE models.

Background: Sample sizes for magnetic resonance imaging (MRI)-based clinical trials in multiple sclerosis (MS) generally assume that lesion counts are reasonably described by the negative binomial (NB) model. Objective: This study aimed to assess the appropriateness of the NB model for lesion count data and to provide sample sizes for placebo-controlled, MRI-based clinical trials in relapsing–remitting MS using a more realistic model. Methods: The fit of the NB model in each arm of five MS clinical trials was assessed using Pearson’s chi-squared statistic. Required sample sizes associated with various tests of treatment effect were estimated by simulating data from a new, longitudinal model for repeated lesion count data on individual patients. Results: Evidence (p < 0.05) against the NB model was found in at least one arm of four of the five trials. If a trial is designed using this model but the resulting clinical data do not follow its assumptions then this trial can be seriously under-powered for assessing differences in mean lesion counts. Conclusion: Sample sizes based on the longitudinal model are more realistic and often smaller than those previously reported using the NB model.

Parking violations cause numerous problems, thus affecting daily mobility. Nevertheless, there are no extensive statistics on illegal parking in Germany, implying that the causes of this misconduct are still unexplored. The objective of this paper is to present a count data modeling approach for parking violations based on video footage taken from the windshields of driving vehicles, incorporating spatial data from OpenStreetMap (OSM). The main benefit of this data source is the transferability of the data collection procedure just by installing a recording device in vehicles of municipal services like waste collection. Moreover, the data from OSM are freely available for all cities. To account for excess zero counts in the street segments, a zero-inflated negative binomial distribution model was used to explain the number of violations per 100 m. “Excess” zeros were modeled using the logit part of the model, whereas the remaining counts of parking violations were fitted by the negative binomial model. Much effort was made to present the results of the count data models in an interpretable way. The most intuitive way seemed to be predictions of parking violations per 100 m (incidence rate) for different settings. Incidence rates were predicted for variations in explanatory variables, holding all other variables constant. We found parking violations per 100 m to be highest in main shopping streets. In addition, free parking spaces negatively- and the number of points of interest (such as buildings, craft stores, and shops) positively affected illegal parking.

This study was aimed at examining the performance of count data models under various outliers and zero inflation situations with simulated data. Poisson, Negative Binomial, Zero-inflated Poisson, Zero-inflated Negative Binomial, Poisson Hurdle and Negative Binomial Hurdle models were considered to test how well each of the model fits the selected datasets having outliers and excess zeros. We found that Zero-inflated Negative Binomial and Negative Binomial Hurdle models were found to be more successful than other count data models. Also the results indicated that in some scenarios, the Negative Binomial model outperformed other models in the presence of outliers and/or excess zeros.

Genetic analysis of fertility in dairy cattle using negative binomial mixed models.

The arrival and departure distribution pattern of Trans Jogja bus passenger is one of the fundamental model for simulation. The purpose of this paper is to build models of passengers flows. This research used passengers data from January to May 2014. There is no policy that change the operation system affecting the nature of this pattern nowadays. The roads, buses, land uses, schedule, and people are relatively still the same. The data then categorized based on the direction, days, and location. Moreover, each category was fitted into some well-known discrete distributions. Those distributions are compared based on its AIC value and BIC. The chosen distribution model has the smallest AIC and BIC value and the negative binomial distribution found has the smallest AIC and BIC value. Probability mass function (PMF) plots of those models were compared to draw generic model from each categorical negative binomial distribution models. The value of accepted generic negative binomial distribution is 0.7064 and 1.4504 of mu. The minimum and maximum passenger vector value of distribution are is 0 and 41.

Background: Parametric count distributions customarily used in demography – the Poisson and negative binomial models – do not offer satisfactory descriptions of empirical distributions of completed cohort parity. One reason is that they cannot model variance-to-mean ratios below unity, i.e., underdispersion, which is typical of low-fertility parity distributions. Statisticians have recently revived two generalised count distributions that can model both over- and underdispersion, but they have not attracted demographers’ attention to date. Objective: The objective of this paper is to assess the utility of these alternative general count distributions, namely the Conway-Maxwell-Poisson and gamma count models, for the modeling of distributions of completed parity. Methods: Simulations and maximum-likelihood estimation are used to assess their fit to empirical data from the Human Fertility Database (HFD). Results: The results show that the generalised count distributions offer a dramatically improved fit compared to customary Poisson and negative binomial models in the presence of under- dispersion, without performance loss in the case of equidispersion or overdispersion. Conclusions: This gain in accuracy suggests generalised count distributions should be used as a matter of course in studies of fertility that examine completed parity as an outcome. Contribution: This note performs a transfer of the state of the art in count data modelling and regression in the more technical statistical literature to the field of demography, allowing demographers to benefit from more accurate estimation in fertility research.

This paper deals with the econometrics of car accidents, that is, the estimation of the relative importance or significance of the factors explaining the number of accidents in a given period on an individual basis. The number of car accidents is a discrete variable and, therefore, represents a count process: the dependent variable takes only nonnegative integer values. Hence, the observed dependent variable is the number of accidents an individual i had in the period considered. The individual characteristics are considered exogenous or predetermined and may or may not be significant factors in explaining the number of accidents. We have estimated four categorical models (linear probability, probit, logit, and multinomial logit) and four count data models (Poisson and negative binomial models with and without individual characteristics in the regression component). It is difficult to compare the econometric results of the different models since some of these models are not nested. However, it is shown that the negative binomial model with a regression component produces a reasonable approximation of the true distribution of accidents. Different statistical tests reject the Poisson models (with and without a regression component) and the negative binomial model without individual characteristics. It is also observed that all estimated models provide the same qualitative results (essentially the same significant variables), but differ when predictions of either the probabilities of accident or the expected number of accidents were made. For quantitative predictions, it is important to select the appropriate model. Moreover, it is shown that, in all models, the individual’s past driving experience is a good predictor of risk. Finally, we apply the statistical results to a model of insurance rating in the presence of moral hazard.

A major focus for transportation safety analysts is the development of crash prediction models, a task for which an extremely wide selection of model types is available. Perhaps the most common crash prediction model is the negative binomial (NB) regression model. The NB model gained popularity due to its relative ease of implementation and its ability to handle overdispersion in crash data. Recently, many new models including the Poisson-Inverse-Gaussian, Sichel, Poisson-Lognormal, and Poisson-Weibull models have been introduced as they can also accommodate overdispersion and could potentially replace the NB model, because many have been found to perform better. All five of the aforementioned models, including the NB model, can be classified as mixed-Poisson models. A mixed-Poisson model arises when an error term, following a chosen mixture distribution, enters the functional form for the Poisson parameter. For the NB model, the mixture distribution is selected as gamma, hence the alternate model name of Poisson-Gamma model. In this paper, confidence intervals (CIs) for the Poisson mean () as well as prediction intervals (PIs) for the Poisson parameter ( alternately referred to as the safety), and the predicted number of crashes at a new site () are derived for each of the aforementioned types of mixed-Poisson models. After the derivations, the theory is put into practice when CIs and PIs are estimated for mixed-Poisson models developed from an animal-vehicle collision data set. Ultimately, this study provides safety analysts with tools to express levels of uncertainty associated with estimates from safety-modeling efforts instead of simply providing point estimates.