A k-Inflated Negative Binomial Mixture Regression Model: Application to Rate–Making Systems

Abstract This article introduces a k-Inflated Negative Binomial mixture distribution/regression model as a more flexible alternative to zero-inflated Poisson distribution/regression model. An EM algorithm has been employed to estimate the model’s parameters. Then, such new model along with a Pareto mixture model have employed to design an optimal rate–making system. Namely, this article employs number/size of reported claims of Iranian third party insurance dataset. Then, it employs the k-Inflated Negative Binomial mixture distribution/regression model as well as other well developed counting models along with a Pareto mixture model to model frequency/severity of reported claims in Iranian third party insurance dataset. Such numerical illustration shows that: (1) the k-Inflated Negative Binomial mixture models provide more fair rate/pure premiums for policyholders under a rate–making system; and (2) in the situation that number of reported claims uniformly distributed in past experience of a policyholder (for instance $k_1=1$ and $k_2=1$ instead of $k_1=0$ and $k_2=2$). The rate/pure premium under the k-Inflated Negative Binomial mixture models are more appealing and acceptable.

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Goodness of fit checks for binomial N-mixture models

10.1101/194340 ◽

2017 ◽

Cited By ~ 1

Author(s):

Jonas Knape ◽

Debora Arlt ◽

Frédéric Barraquand ◽

Åke Berg ◽

Mathieu Chevalier ◽

...

Keyword(s):

Mixture Models ◽

Mixture Model ◽

Binomial Distribution ◽

Detection Probability ◽

Goodness Of Fit ◽

Negative Binomial ◽

List Type ◽

Computationally Efficient ◽

Abundance Distribution ◽

Lack Of Fit

AbstractBinomial N-mixture models are commonly applied to analyze population survey data. By estimating detection probabilities, N-mixture models aim at extracting information about abundances in terms of actual and not just relative numbers. This separation of detection probability and abundance relies on parametric assumptions about the distribution of individuals among sites and of detections of individuals among repeat visits to sites. Current methods for checking assumptions are limited, and their computational complexity have hindered evaluations of their performances.We develop computationally efficient graphical goodness of fit checks and measures of overdispersion for binomial N-mixture models. These checks are illustrated in a case study, and evaluated in simulations under two scenarios. The two scenarios assume overdispersion in the abundance distribution via a negative binomial distribution or in the detection probability via a beta-binomial distribution. We evaluate the ability of the checks to detect lack of fit, and how lack of fit affects estimates of abundances.The simulations show that if the parametric assumptions are incorrect there can be severe biases in estimated abundances: negatively if there is overdispersion in abundance relative to the fitted model and positively if there is overdispersion in detection. Our goodness of fit checks performed well in detecting lack of fit when the abundance distribution is overdispersed, but struggled to detect lack of fit when detections were overdispersed. We show that the inability to detect lack of fit due to overdispersed detection is caused by a fundamental similarity between N-mixture models with beta-binomial detections and N-mixture models with negative binomial abundances.The strong biases in estimated abundances that can occur in the binomial N-mixture model when the distribution of individuals among sites, or the detection model, is mis-specified implies that checking goodness of fit is essential for sound inference in ecological studies that use these methods. To check the assumptions we provide computationally efficient goodness of fit checks that are available in an R-package nmixgof. However, even when a binomial N-mixture model appears to fit the data well, estimates are not robust in the presence of overdispersion unless additional information about detection is collected.

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Bayesian negative binomial mixture regression models for the analysis of sequence count and methylation data

Biometrics ◽

10.1111/biom.12962 ◽

2018 ◽

Vol 75 (1) ◽

pp. 183-192 ◽

Cited By ~ 3

Author(s):

Qiwei Li ◽

Alberto Cassese ◽

Michele Guindani ◽

Marina Vannucci

Keyword(s):

Regression Models ◽

Negative Binomial ◽

Methylation Data ◽

Mixture Regression ◽

Binomial Mixture ◽

Mixture Regression Models

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Predicting tree recruitment with negative binomial mixture models

Forest Ecology and Management ◽

10.1016/j.foreco.2012.01.028 ◽

2012 ◽

Vol 270 ◽

pp. 209-215 ◽

Cited By ~ 17

Author(s):

Xiongqing Zhang ◽

Yuancai Lei ◽

Daoxiong Cai ◽

Fengqiang Liu

Keyword(s):

Mixture Models ◽

Negative Binomial ◽

Tree Recruitment ◽

Binomial Mixture

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OPTIMAL BONUS-MALUS SYSTEMS USING FINITE MIXTURE MODELS

Astin Bulletin ◽

10.1017/asb.2013.31 ◽

2014 ◽

Vol 44 (2) ◽

pp. 417-444 ◽

Cited By ~ 19

Author(s):

George Tzougas ◽

Spyridon Vrontos ◽

Nicholas Frangos

Keyword(s):

Mixture Models ◽

Posterior Probability ◽

Negative Binomial ◽

Finite Mixture Models ◽

A Priori ◽

Frequency Component ◽

Finite Mixture ◽

Automobile Insurance ◽

A Posteriori ◽

Binomial Mixture

AbstractThis paper presents the design of optimal Bonus-Malus Systems using finite mixture models, extending the work of Lemaire (1995; Lemaire, J. (1995) Bonus-Malus Systems in Automobile Insurance. Norwell, MA: Kluwer) and Frangos and Vrontos (2001; Frangos, N. and Vrontos, S. (2001) Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin, 31(1), 1–22). Specifically, for the frequency component we employ finite Poisson, Delaporte and Negative Binomial mixtures, while for the severity component we employ finite Exponential, Gamma, Weibull and Generalized Beta Type II mixtures, updating the posterior probability. We also consider the case of a finite Negative Binomial mixture and a finite Pareto mixture updating the posterior mean. The generalized Bonus-Malus Systems we propose, integrate risk classification and experience rating by taking into account both the a priori and a posteriori characteristics of each policyholder.

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Generalized ridge estimator in negative binomial regression model

Journal of Physics Conference Series ◽

10.1088/1742-6596/1897/1/012019 ◽

2021 ◽

Vol 1897 (1) ◽

pp. 012019

Author(s):

Nadwa Khazaal Rashad ◽

Nawal Mahmood Hammood ◽

Zakariya Yahya Algamal

Keyword(s):

Regression Model ◽

Negative Binomial ◽

Negative Binomial Regression ◽

Negative Binomial Regression Model ◽

Ridge Estimator ◽

Binomial Regression ◽

Generalized Ridge Estimator

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Mixture models as a method for comparative sociality: social networks and demographic change in resident killer whales

Behavioral Ecology and Sociobiology ◽

10.1007/s00265-021-03006-3 ◽

2021 ◽

Vol 75 (4) ◽

Author(s):

Samuel Ellis ◽

Daniel W. Franks ◽

Michael N. Weiss ◽

Michael A. Cant ◽

Paolo Domenici ◽

...

Keyword(s):

Mixture Models ◽

Demographic Change ◽

Social Preferences ◽

Social Bonds ◽

Killer Whales ◽

Animal Populations ◽

Binomial Mixture ◽

Opposite Sex ◽

Social Associations ◽

Social Association

Abstract In studies of social behaviour, social bonds are usually inferred from rates of interaction or association. This approach has revealed many important insights into the proximate formation and ultimate function of animal social structures. However, it remains challenging to compare social structure between systems or time-points because extrinsic factors, such as sampling methodology, can also influence the observed rate of association. As a consequence of these methodological challenges, it is difficult to analyse how patterns of social association change with demographic processes, such as the death of key social partners. Here we develop and illustrate the use of binomial mixture models to quantitatively compare patterns of social association between networks. We then use this method to investigate how patterns of social preferences in killer whales respond to demographic change. Resident killer whales are bisexually philopatric, and both sexes stay in close association with their mother in adulthood. We show that mothers and daughters show reduced social association after the birth of the daughter’s first offspring, but not after the birth of an offspring to the mother. We also show that whales whose mother is dead associate more with their opposite sex siblings and with their grandmother than whales whose mother is alive. Our work demonstrates the utility of using mixture models to compare social preferences between networks and between species. We also highlight other potential uses of this method such as to identify strong social bonds in animal populations. Significance statement Comparing patters of social associations between systems, or between the same systems at different times, is challenging due to the confounding effects of sampling and methodological differences. Here we present a method to allow social associations to be robustly classified and then compared between networks using binomial mixture models. We illustrate this method by showing how killer whales change their patterns of social association in response to the birth of calves and the death of their mother. We show that after the birth of her calf, females associate less with their mother. We also show that whales’ whose mother is dead associate more with their opposite sex siblings and grandmothers than whales’ whose mother is alive. This clearly demonstrates how this method can be used to examine fine scale temporal processes in animal social systems.

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Statistical models for analyzing count data: predictors of length of stay among HIV patients in Portugal using a multilevel model

BMC Health Services Research ◽

10.1186/s12913-021-06389-1 ◽

2021 ◽

Vol 21 (1) ◽

Author(s):

Ahmed Nabil Shaaban ◽

Bárbara Peleteiro ◽

Maria Rosario O. Martins

Keyword(s):

Length Of Stay ◽

Regression Model ◽

Random Effects ◽

Count Data ◽

Negative Binomial ◽

Negative Binomial Regression ◽

Comprehensive Approach ◽

Negative Binomial Regression Model ◽

Hiv Patients ◽

Binomial Regression

Abstract Background This study offers a comprehensive approach to precisely analyze the complexly distributed length of stay among HIV admissions in Portugal. Objective To provide an illustration of statistical techniques for analysing count data using longitudinal predictors of length of stay among HIV hospitalizations in Portugal. Method Registered discharges in the Portuguese National Health Service (NHS) facilities Between January 2009 and December 2017, a total of 26,505 classified under Major Diagnostic Category (MDC) created for patients with HIV infection, with HIV/AIDS as a main or secondary cause of admission, were used to predict length of stay among HIV hospitalizations in Portugal. Several strategies were applied to select the best count fit model that includes the Poisson regression model, zero-inflated Poisson, the negative binomial regression model, and zero-inflated negative binomial regression model. A random hospital effects term has been incorporated into the negative binomial model to examine the dependence between observations within the same hospital. A multivariable analysis has been performed to assess the effect of covariates on length of stay. Results The median length of stay in our study was 11 days (interquartile range: 6–22). Statistical comparisons among the count models revealed that the random-effects negative binomial models provided the best fit with observed data. Admissions among males or admissions associated with TB infection, pneumocystis, cytomegalovirus, candidiasis, toxoplasmosis, or mycobacterium disease exhibit a highly significant increase in length of stay. Perfect trends were observed in which a higher number of diagnoses or procedures lead to significantly higher length of stay. The random-effects term included in our model and refers to unexplained factors specific to each hospital revealed obvious differences in quality among the hospitals included in our study. Conclusions This study provides a comprehensive approach to address unique problems associated with the prediction of length of stay among HIV patients in Portugal.

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