Poisson size-biased Lindley distribution and its applications

In this paper, a new model for count data is introduced by compounding the Poisson distribution with size-biased three-parameter Lindley distribution. Statistical properties, such as reliability, hazard rate, reverse hazard rate, Mills ratio, moments, shewness, kurtosis, moment genrating function, probability generating function and order statistics, have been discussed. Moreover, the collective risk model is discussed by considering the proposed distrubution as the primary distribution and the expoential and Erlang distributions as the secondary ones. Parameter estimation is done using maximum likelihood estimation (MLE). Finally a real dataset is discussed to demonstrate the suitability and applicability of the proposed distribution in modeling count dataset.

ON THE FAMILY OF DISTRIBUTIONS WITH ACTUARIAL APPLICATIONS

ABSTRACT In this paper, a new three-parameter discrete family of distributions, the $$r{\cal B}ell$$ family, is introduced. The family is based on series expansion of the r-Bell polynomials. The proposed model generalises the classical Poisson and the recently proposed Bell and Bell–Touchard distributions. It exhibits interesting stochastic properties. Its probabilities can be computed by a recursive formula that allows us to calculate the probability function of the amount of aggregate claims in the collective risk model in terms of an integral equation. Univariate and bivariate regression models are presented. The former regression model is used to explain the number of out-of-use claims in an automobile insurance portfolio, by showing a good out-of-sample performance. The latter is used to describe the number of out-of-use and parking claims jointly. This family provides an alternative to other traditionally used distributions to describe count data such as the negative binomial and Poisson-inverse Gaussian models.

A Desirable Aspect in the Variance Premium in a Collective Risk Model

This paper focuses on the study of the Collective and Bayes Premiums, under the Variance Premium Principle, in the classic Collective Risk Poisson-Exponential Model. A bivariate prior distribution is considered for both the parameter of the distribution of the number of claims and that of the distribution of the claim amount, assuming independence between these parameters. Furthermore, we analyze the consequences on these premiums of small levels of contamination in the structure functions, and find that the premiums are not sensitive to small levels of uncertainty. These results extend the conclusions obtained in Gómez-Déniz et al. (2000), where only variations in the parameter for the number of claims and its effects on premiums were studied.

Frequency and Severity Dependence in the Collective Risk Model: An Approach Based on Sarmanov Distribution

In actuarial mathematics, the claims of an insurance portfolio are often modeled using the collective risk model, which consists of a random number of claims of independent, identically distributed (i.i.d.) random variables (r.v.s) that represent cost per claim. To facilitate computations, there is a classical assumption of independence between the random number of such random variables (i.e., the claims frequency) and the random variables themselves (i.e., the claim severities). However, recent studies showed that, in practice, this assumption does not always hold, hence, introducing dependence in the collective model becomes a necessity. In this sense, one trend consists of assuming dependence between the number of claims and their average severity. Alternatively, we can consider heterogeneity between the individual cost of claims associated with a given number of claims. Using the Sarmanov distribution, in this paper we aim at introducing dependence between the number of claims and the individual claim severities. As marginal models, we use the Poisson and Negative Binomial (NB) distributions for the number of claims, and the Gamma and Lognormal distributions for the cost of claims. The maximum likelihood estimation of the proposed Sarmanov distribution is discussed. We present a numerical study using a real data set from a Spanish insurance portfolio.

Multivariate Collective Risk Model: Dependent Claim Numbers and Panjer’s Recursion

In this paper, we discuss a generalization of the collective risk model and of Panjer’s recursion. The model we consider consists of several business lines with dependent claim numbers. The distributions of the claim numbers are assumed to be Poisson mixture distributions. We let the claim causes have certain dependence structures and prove that Panjer’s recursion is also applicable by finding an appropriate equivalent representation of the claim numbers. These dependence structures are of a stochastic non-negative linear nature and may also produce negative correlations between the claim causes. The consideration of risk groups also includes dependence between claim sizes. Compounding the claim causes by common distributions also keeps Panjer’s recursion applicable.

A zero truncated discrete distribution: Theory and applications to count data

The analysis and modeling of zero truncated count data is of primary interest in many elds such as engineering, public health, sociology, psychology, epidemiology. Therefore, in this article we have proposed a new and simple structure model, named a zero truncated discrete Lindley distribution. Thedistribution contains some submodels and represents a two-component mixture of a zero truncated geometric distribution and a zero truncated negative binomial distribution with certain parameters. Several properties of the distribution are obtained such as mean residual life function, probability generating function, factorial moments, negative moments, moments of residual life function, Bonferroni and Lorenz curves, estimation of parameters, Shannon and Renyi entropies, order statistics with the asymptotic distribution of their extremes and range, a characterization, stochastic ordering and stress-strength parameter. Moreover, the collective risk model is discussed by considering theproposed distribution as primary distribution and exponential and Erlang distributions as secondary ones. Test and evaluation statistics as well as three real data applications are considered to assess the peformance of the distribution among the most frequently zero truncated discrete probability models.