A new look at the inverse Gaussian distribution with applications to insurance and economic data

Abstract

Insurance and economic data are often positive, and we need to take into account this peculiarity in choosing a statistical model for their distribution. An example is the inverse Gaussian (IG), which is one of the most famous and considered distributions with positive support. With the aim of increasing the use of the IG distribution on insurance and economic data, we propose a convenient mode-based parameterization yielding the reparametrized IG (rIG) distribution; it allows/simplifies the use of the IG distribution in various branches of statistics, and we give some examples. In nonparametric statistics, we define a smoother based on rIG kernels. By construction, the estimator is well-defined and does not allocate probability mass to unrealistic negative values. We adopt likelihood cross-validation to select the smoothing parameter. In robust statistics, we propose the contaminated IG distribution, a heavy-tailed generalization of the rIG distribution to accommodate mild outliers. Finally, for modelbased clustering and semiparametric density estimation, we present finite mixtures of rIG distributions. We use the EM algorithm to obtain maximum likelihood estimates of the parameters of the mixture and contaminated models. We use insurance data about bodily injury claims, and economic data about incomes of Italian households, to illustrate the models.

New methods to define heavy-tailed distributions with applications to insurance data

“…The new developments have been made through many different approaches such as (i) transformation of variables, (ii) composition of two or more distributions, (iii) compounding of distributions and (iv) finite mixture of distributions. Recent studies of Eling [5] and Adcock et al [6] identify that skew-normal and skew student t distributions are the best competitors as the skewed distributions adjust right-skewness and high kurtosis; for further detail see, Shushi [7] and Punzo [8]. However, insurance losses and financial risks take values on the positive real line and hence these skew class of distributions may not be appropriate as they are defined on R. In such situations, the transformation of variables, particularly the exponential transformation, has proven to be substantial; see, for example, Azzalini et al [9].…”

“…Another prominent approach is compounding of distributions to cater data modelling with unimodality, right-skewness and heavy tails [8,15,16]. However, the density obtained via this method may not have a closed form expression which makes the estimation more cumbersome as shown in Punzo et al [15].…”

The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial Data

“…There are some methods that have been introduced to construct new distributions with heavier tails than the exponential distribution, called the transformation method, compounding of distributions, composition of two or more models and finite mixture distributions. The interested reader can refer to Eling [11], Kazemi and Noorizadeh [12], Bakar et al [13], Punzo [14], Mazza and Punzo [15], Miljkovic and Grun [16], and Punzo et al [17].…”

Type-I heavy tailed family with applications in medicine, engineering and insurance

“…Recent the study of [ 9 ] showed that skewed student t model and skewed-normal model are the best competitors as the skewed distributions adjust right-skewness and high kurtosis; for further detail see [ 10 ]. Financial risks and the insurance losses take positive values on the real line and consequently these skew models may not be suitable choice.…”