Objective Bayesian modelling of insurance risks with the skewed Student‐t distribution

Abstract

Insurance risks data typically exhibit skewed behaviour. In this paper, we propose a Bayesian approach to capture the main features of these data sets. This work extends a methodology recently introduced in the literature by considering an extra parameter that captures the skewness of the data. In particular, a skewed Student‐t distribution is considered. Two data sets are analysed: the Danish fire losses and the US indemnity loss. The analysis is carried with an objective Bayesian approach. For the discrete parameter representing the number of the degrees of freedom, we adopt a novel prior recently appeared in the literature. Copyright © 2017 John Wiley & Sons, Ltd.

Objective priors for the number of degrees of freedom of a multivariate t distribution and the t-copula

“…However, given that the proposed prior does not depend on the choice of the marginals, it is possible to employ more flexible marginal distributions, such as the two-piece Student-t (see Rubio and Steel, 2015 for an extensive discussion of the family of two-piece distributions), in order to capture skewness and heavy tails. Leisen et al (2017) proposed an objective prior for the degrees of freedom parameter in the univariate two-piece Student-t distribution, which is constructed using the loss-based principle discussed in Section 3. They show that this prior does not depend on the skewness parameter, and that it coincides with that proposed in Villa and Walker (2014) for the univariate Student-t distribution (see Section 3).…”

Loss-based approach to two-piece location-scale distributions with applications to dependent data

“…is the Kullback-Leibler divergence. Following Leisen et al (2017), we introduce a theorem (which proof is in Appendix A) to study the form of the Kullback-Leibler divergence and consequently of the loss-based prior for the tail parameter p.…”

“…The SST has some special cases: if α = 1/2, it is the usual Student-t with p degrees of freedom; if p = 1, is the skewed Cauchy, while for p → ∞, it converges to the skewed normal distribution. Leisen et al (2017) have proposed a loss-based prior for the tail parameter p of the SST distribution. Therefore, hereafter we will give limited attention to this distribution and we will focus on the remaining two distributions.…”

“…The performance of the proposed approach is illustrated through simulation experiments and real data analysis. The methodology yields improvements in density forecasts, as shown by the analysis we carry out on the electricity prices in Nordpool markets.MSC: 62F15, 62M10.In the literature, the asymmetric Laplace distribution (ALD) or the asymmetric Studentt distribution (AST) have gained importance in a wide range of disciplines, such as economics (Zhao et al, 2007;Leisen et al, 2017), financial analysis (Zhu and Galbraith, 2010;Kozubowski and Podgorski, 2001;Harvey and Lange, 2016) and microbiology (Rubio and Steel, 2011). However, the application of the SEPD and SGLD to represent the errors of time series and regression models, has received limited attention in the context of objective Bayesian analysis.…”

Objective Bayesian analysis for the multivariate skew-t model

“…In the paper we will concentrate on the Azzalini-type skew-t distribution. For a Bayesian analysis of the "two-piece" t distribution one can refer to Rubio et al (2015) and Leisen et al (2016) where a new objective prior is introduced for the degrees of freedom parameter. Following Azzalini & Capitanio (2014), their version of the multivariate skewt distribution can be obtained as a scale mixture of multivariate skew-normal distributions.…”