scholarly journals A Generalization of Automobile Insurance Rating Models: The Negative Binomial Distribution with a Regression Component

1989 ◽  
Vol 19 (2) ◽  
pp. 199-212 ◽  
Author(s):  
Georges Dionne ◽  
Charles Vanasse
Keyword(s):  
Negative Binomial Distribution ◽  
Negative Binomial ◽  
A Priori ◽  
Poisson Model ◽  
Random Variable ◽  
Second Step ◽  
Automobile Insurance ◽  
Binomial Model ◽  
A Posteriori ◽  
Available Information

AbstractThe objective of this paper is to provide an extension of well-known models of tarification in automobile insurance. The analysis begins by introducing a regression component in the Poisson model in order to use all available information in the estimation of the distribution. In a second step, a random variable is included in the regression component of the Poisson model and a negative binomial model with a regression component is derived. We then present our main contribution by proposing a bonus-malus system which integrates a priori and a posteriori information on an individual basis. We show how net premium tables can be derived from the model. Examples of tables are presented.

scholarly journals OPTIMAL BONUS-MALUS SYSTEMS USING FINITE MIXTURE MODELS

2014 ◽  
Vol 44 (2) ◽  
pp. 417-444 ◽  
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.

Ehrenberg's Negative Binomial Model Applied to Grocery Store Trips

1980 ◽  
Vol 17 (3) ◽  
pp. 385-390 ◽  
Author(s):  
Gil A. Frisbie
Keyword(s):  
Negative Binomial Distribution ◽  
Goodness Of Fit ◽  
Negative Binomial ◽  
Grocery Store ◽  
Distribution Model ◽  
Grocery Stores ◽  
Good Representation ◽  
Binomial Model ◽  
Food Stores ◽  
Goodness Of Fit Tests

Ehrenberg's negative binomial distribution model is applied to a new facet of consumer behavior, the frequency of household filler trips to food stores. Goodness-of-fit tests and intertemporal predictions are assessed. The overall verdict is that the model serves as a good representation of the trips to grocery stores.

Estimating Rates of Recurrent Peritonitis for Patients on CAPO

1985 ◽  
Vol 5 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Edward F. Vonesh
Keyword(s):  
Risk Factors ◽  
Medical Devices ◽  
Negative Binomial ◽  
Survival Curve ◽  
Probability Model ◽  
Major Complication ◽  
Poisson Model ◽  
Negative Binomial Model ◽  
Binomial Model ◽  
The Mean

Recurrent peritonitis is a major complication of Continuous Ambulatory Peritoneal Dialysis (CAPD). As a therapy for patients with end stage renal disease, CAPD entails a continuous interaction between patient and various medical devices. The assumptions one makes regarding this interaction play an essential role when estimating the rate of recurrent peritonitis for a given patient population. Assuming that each patient has a constant rate of peritonitis, two models for evaluating the risk of recurrent peritonitis are considered. One model, the Poisson probability model, applies when the rate of peritonitis is the same from patient to patient. When this occurs, the frequency of peritoneal infections will be randomly distributed among patients (Corey, 1981). A second model, the negative binomial probability model, applies when the rate of peritonitis varies from one patient to another. In this event, the distribution of peritoneal infections will differ from patient to patient. The poisson model would be applicable when, for example, patients behave similarly with respect to their interactions with the medical devices and with potential risk factors. The negative binomial model, on the other hand, makes allowances for patient differences both in terms of their handling of routine exchanges and in their exposure to various risk factors. This paper provides methods for estimating the mean peritonitis rate under each model. In addition, “survival” curve estimates depicting the probability of remaining peritonitis free (i.e. “surviving”) over time are provided. It is shown, using data from a multi-center clinical trial, that the risk of peritonitis is best described in terms of survival curves rather than the mean peritonitis rate. For both models, the mean peritonitis rate was found to be 0.85 episodes per year. However, under the negative binomial model, the one-year survival rate, expressed as the percentage of patients remaining free of peritonitis, is 52% as compared with only 42% under the Poisson model. Moreover, the negative binomial model provided a significantly better fit to the observed frequency of peritonitis. These findings suggest that the negative binomial model provides a more realistic and accurate portrayal of the risk of peritonitis and that this risk is not nearly as high as would otherwise be indicated by a Poisson analysis.

scholarly journals Modeling Hospitalization Decision and Utilization for the Elderly in China

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Xin Xu ◽  
Dongxiao Chu
Keyword(s):  
Negative Binomial ◽  
Hospital Readmissions ◽  
Poisson Model ◽  
Medical Insurance ◽  
The Elderly ◽  
Negative Binomial Model ◽  
Binomial Model ◽  
Factors Affecting ◽  
Basic Medical Insurance ◽  
Basic Medical

Getting medical services has become more difficult and expensive in China, which led to a problem of illness not being treated and a large number of zeros in the statistics of being hospitalized for the elderly. Traditional classic models such as the Poisson model and the negative binomial model cannot fit this kind of data well. One aim of this study was to use zero-inflated and hurdle models to better solve the problem of excess zeros. Another aim was to discover the factors affecting the decision-making behavior of the elderly being hospitalized and hospitalization service utilization. Therefore, the XGBoost model was firstly introduced to rank the importance of influencing factors in this paper. It was found that the zero-inflated negative binomial model performed best. The results showed that the elderly who had enjoyed NRCM or ERBMI/URBMI were more likely to have a higher number of hospitalizations. This indicated that the high cost of hospitalization had prevented the willingness of the elderly being hospitalized, but the basic medical insurance had increased the times of their repeated hospital readmissions. Policy efforts should be made to improve the level of basic medical insurance.

A unified approach to limit theorems for urn models

1979 ◽  
Vol 16 (01) ◽  
pp. 154-162 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Limit Theorem ◽  
Characteristic Function ◽  
Negative Binomial Distribution ◽  
Limit Theorems ◽  
Negative Binomial ◽  
Random Variable ◽  
Unified Approach ◽  
Special Cases ◽  
Occupancy Problems ◽  
General Method

An urn contains A balls of each of N colours. At random n balls are drawn in succession without replacement, with replacement or with replacement together with S new balls of the same colour. Let Xk be the number of drawn balls having colour k, k = 1, …, N. For a given function f the characteristic function of the random variable ZM = f(X 1)+ … + f(XM ), M ≦ N, is derived. A limit theorem for ZM when M, N, n → ∞is proved by a general method. The theorem covers many special cases discussed separately in the literature. As applications of the theorem limit distributions are obtained for some occupancy problems and for dispersion statistics for the binomial, Poisson and negative-binomial distribution.

A POSTERIORI RATEMAKING WITH PANEL DATA

2014 ◽  
Vol 44 (3) ◽  
pp. 587-612 ◽  
Author(s):  
Jean-Philippe Boucher ◽  
Rofick Inoussa
Keyword(s):  
Panel Data ◽  
A Priori ◽  
Automobile Insurance ◽  
Insurance Company ◽  
A Posteriori ◽  
Numerical Illustration ◽  
Section Data ◽  
The Past ◽  
Car Insurance ◽  
Insurance Data

AbstractRatemaking is one of the most important tasks of non-life actuaries. Usually, the ratemaking process is done in two steps. In the first step, a priori ratemaking, an a priori premium is computed based on the characteristics of the insureds. In the second step, called the a posteriori ratemaking, the past claims experience of each insured is considered to the a priori premium and set the final net premium. In practice, for automobile insurance, this correction is usually done with bonus-malus systems, or variations on them, which offer many advantages. In recent years, insurers have accumulated longitudinal information on their policyholders, and actuaries can now use many years of informations for a single insured. For this kind of data, called panel or longitudinal data, we propose an alternative to the two-step ratemaking approach and argue this old approach should no longer be used. As opposed to a posteriori models of cross-section data, the models proposed in this paper generate premiums based on empirical results rather than inductive probability. We propose a new way to deal with bonus-malus systems when panel data are available. Using car insurance data, a numerical illustration using at-fault and non-at-fault claims of a Canadian insurance company is included to support this discussion. Even if we apply the model for car insurance, as long as another line of business uses past claim experience to set the premiums, we maintain that a similar approach to the model proposed should be used.

A characterization of the gamma distribution by the negative binomial distribution

1980 ◽  
Vol 17 (04) ◽  
pp. 1138-1144 ◽  
Author(s):  
Jan Engel ◽  
Mynt Zijlstra
Keyword(s):  
Poisson Process ◽  
Exponential Distribution ◽  
Gamma Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Random Variable ◽  
One To One

It is proved that for a Poisson process there exists a one-to-one relation between the distribution of the random variable N(Y) and the distribution of the non-negative random variable Y. This relation is used to characterize the gamma distribution by the negative binomial distribution. Furthermore it is applied to obtain some characterizations of the exponential distribution.

scholarly journals Leaf count overdispersion in coffee seedlings

2019 ◽  
Vol 49 (4) ◽  
Author(s):  
Edilson Marcelino Silva ◽  
Thais Destefani Ribeiro Furtado ◽  
Jaqueline Gonçalves Fernandes ◽  
Marcelo Ângelo Cirillo ◽  
Joel Augusto Muniz
Keyword(s):  
Count Data ◽  
Negative Binomial ◽  
Block Design ◽  
Poisson Model ◽  
Relevant Information ◽  
Relative Increase ◽  
Negative Binomial Model ◽  
Binomial Model ◽  
Coffee Bean ◽  
Number Of Leaves

ABSTRACT: Coffee crops play an important role in Brazilian agriculture, with a high level of social and economic participation resulting from the jobs created in the supply chain and from the income obtained by producers and the revenue generated for the country from coffee bean export. In coffee plant growth, leaves have a determinant role in higher production; therefore, the leaf count per plant provides relevant information to producers for adequate crop management, such as foliar fertilizer applications. To describe count data, the Poisson model is the most commonly employed model; when count data show overdispersion, the negative binomial model has been determined to be more adequate. The objective of this study was to compare the fitness of the Poisson and negative binomial models to data on the leaf count per plant in coffee seedlings. Data were collected from an experiment with a randomized block design with 30 treatments and three replicates and four plants per plot. Data from only one treatment, in which the number of leaves was counted over time, were employed. The first count was conducted on 8 April 2016, and the other counts were performed 18, 32, 47, 62, 76, 95, 116, 133, and 153 days after the first evaluation, for a total of ten measurements. The fitness of the models was assessed based on deviance values and simulated envelopes for residuals. Results of fitness assessment indicated that the Poisson model was inadequate for describing the data due to overdispersion. The negative binomial model adequately fitted the observations and was indicated to describe the number of leaves of coffee plants. Based on the negative binomial model, the expected relative increase in the number of leaves was 0.9768% per day.

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