Bayesian Statistics in Actuarial Science

Klugman, Stuart A.

doi:10.1007/978-94-017-0845-6

Cited by 80 publications

(35 citation statements)

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“…In recent y ears, several researchers have utilized Bayesian inference for the NBD (e.g., Deely and Smith 1998Klugman 1992Schl uter, Deely, and Nicholson 1997. However, none of these have obtained closed-form inferences, as we do here.…”

Section: Introductionmentioning

confidence: 70%

Bayesian Inference for the Negative Binomial Distribution via Polynomial Expansions

Bradlow

Hardie

Fader

2002

Journal of Computational and Graphical Statistics

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To date, Bayesian inferences for the negative binomial distribution (NBD) have relied on computationally intensive numerical methods (e.g., Markov chain Monte Carlo) as it is thought that the posterior densities of interest are not amenable to closed-form integration. In this article, we present a "closed-form" solution to the Bayesian inference problem for the NBD that can be written as a sum of polynomial terms. The key insight is to approximate the ratio of two gamma functions using a polynomial expansion, which then allows for the use of a conjugate prior. Given this approximation, we arrive at closed-form expressions for the moments of both the marginal posterior densities and the predictive distribution by integrating the terms of the polynomial expansion in turn (now feasible due to conjugacy). We demonstrate via a large-scale simulation that this approach is very accurate and that the corresponding gains in computing time are quite substantial. Furthermore, even in cases where the computing gains are more modest our approach provides a method for obtaining starting values for other algorithms, and a method for data exploration. AbstractTo date, Bayesian inferences for the negative binomial distribution (NBD) have relied on computationally intensive n umerical methods (e.g., Markov chain Monte Carlo) as it is thought that the posterior densities of interest are not amenable to closed-form integration. In this paper, we present closed-form solutions to the Bayesian inference problem for the NBD. The key insight is to approximate the ratio of two gamma functions using a polynomial expansion, which then allows for the use of a conjugate prior. Given this approximation, we arrive a t closed-form expressions for the moments of both the marginal posterior densities and the predictive distribution. We demonstrate that this approach is very accurate and that the corresponding gains in computing time are quite substantial.

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Section: Introductionmentioning

confidence: 70%

Bayesian Inference for the Negative Binomial Distribution via Polynomial Expansions

Bradlow

Hardie

Fader

2002

Journal of Computational and Graphical Statistics

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“…Studies which acknowledge and account for parameter uncertainty have begun to emerge in recent years. Most of these deal with general insurance rather than life or investment problems (for example : Klugman, 1992;McNeil, 1997;Pai, 1997;Dickson, Tedesco & Zenwirth, 1998;Scollnik, 1998). On the investment front, Harris (1999) uses Markov chain Monte Carlo methods to select a model for price inflation and to investigate parameter uncertainty.…”

Section: Measuring Uncertaintymentioning

confidence: 99%

A discussion of parameter and model uncertainty in insurance

Cairns

2000

Insurance: Mathematics and Economics

123

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In this paper we consider the process of modelling uncertainty. In particular we are concerned with making inferences about some quantity of interest which, at present, has been unobserved. Examples of such a quantity include the probability of ruin of a surplus process, the accumulation of an investment, the level or surplus or deficit in a pension fund and the future volume of new business in an insurance company. Uncertainty in this quantity of interest, y, arises from three sources:• uncertainty due to the stochastic nature of a given model;• uncertainty in the values of the parameters in a given model;• uncertainty in the model underlying what we are able to observe and determining the quantity of interest.It is common in actuarial science to find that the first source of uncertainty is the only one which receives rigorous attention. A limited amount of research in recent years has considered the effect of parameter uncertainty, while there is still considerable scope for development of methods which deal in a balanced way with model risk. Here we discuss a methodology which allows all three sources of uncertainty to be assessed in a more coherent fashion.

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“…Papers using Poisson-Gamma models abound in risk theory literature (see e.g. Eichenauer et al, 1988;Klugman, 1992;Freifelder, 1974, and references therein). From (1), the true individual premium is given by P (θ) = θ e α .…”

Section: The Modelmentioning

confidence: 99%

The Esscher premium principle in risk theory: a Bayesian sensitivity study

Gómez–Déniz

Hernández-Bastida

Polo

1999

Insurance: Mathematics and Economics

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In this paper the Esscher premium calculation principle is applied to the non-compound collective model in a robust Bayesian context. We consider that uncertainty with regard to the prior distribution can be represented by the assumption that the unknown prior distribution belongs to a class of distributions and examine the ranges of the Bayesian premium when the priors belong to such a class. The assessment of the influence of the prior is termed sensitivity analysis or robustness analysis.

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Bayesian Statistics in Actuarial Science

Cited by 80 publications

References 0 publications

Bayesian Inference for the Negative Binomial Distribution via Polynomial Expansions

Bayesian Inference for the Negative Binomial Distribution via Polynomial Expansions

A discussion of parameter and model uncertainty in insurance

The Esscher premium principle in risk theory: a Bayesian sensitivity study

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