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In most of the United States, insurance companies may use gender to determine car insurance rates. In addition, several studies have shown that women over the age of 25 generally pay more than men for car insurance. Then, we investigate whether the distributions of claims for women and men differ in location, scale and shape by means of the GAMLSS regression framework, using microdata provided by U.S. and Australian insurance companies, to use this evidence to support policy makers' decisions. We also develop a parametric-bootstrap test to investigate the tail behavior of the distributions. When covariates are not considered, the distribution of claims does not appear to differ by gender. When covariates are included, the regressions provide mixed evidence for the location parameter. However, for female claimants, the spread of the distribution is lower. Our research suggests that, at least for the contexts analyzed, there is no clear statistical reason for charging higher rates to women. While providing evidence to support unisex insurance pricing policies, given the limitations represented by the use of country-specific data, this paper aims to promote further research on this topic with different datasets to corroborate our findings and draw more general conclusions. [ABSTRACT FROM AUTHOR]

Insurance loss data have peculiar features that can rarely be accounted for by simple parametric distributions. Thus, in this manuscript, we first introduce a new type of location mixture model: the mode mixture. By using convenient mode-parameterized hump-shaped distributions, we present a family of eight mode mixture of unimodal distributions. Then, we fit these models to two real insurance loss datasets, where they are evaluated in terms of goodness of fit and ability to reproduce classical risk measures. We extend the comparisons to existing models based on mode-parameterized hump-shaped distributions. Lastly, using simulated data, we further investigate the performance of the estimated risk measures of our models. [ABSTRACT FROM AUTHOR]

Many statistical problems involve the estimation of a d × d orthogonal matrix Q . Such an estimation is often challenging due to the orthonormality constraints on Q . To cope with this problem, we use the well-known PLU decomposition, which factorizes any invertible d × d matrix as the product of a d × d permutation matrix P , a d × d unit lower triangular matrix L , and a d × d upper triangular matrix U . Thanks to the QR decomposition, we find the formulation of U when the PLU decomposition is applied to Q . We call the result as PLR decomposition; it produces a one-to-one correspondence between Q and the d d - 1 / 2 entries below the diagonal of L , which are advantageously unconstrained real values. Thus, once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to L . For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of Q in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses. [ABSTRACT FROM AUTHOR]

The cluster-weighted model (CWM) is a mixture model with random covariates that allows for flexible clustering/classification and distribution estimation of a random vector composed of a response variable and a set of covariates. Within this class of models, the generalized linear exponential CWM is here introduced especially for modeling bivariate data of mixed-type. Its natural counterpart in the family of latent class models is also defined. Maximum likelihood parameter estimates are derived using the expectation-maximization algorithm and some computational issues are detailed. Through Monte Carlo experiments, the classification performance of the proposed model is compared with other mixture-based approaches, consistency of the estimators of the regression coefficients is evaluated, and several likelihood-based information criteria are compared for selecting the number of mixture components. An application to real data is also finally considered. [ABSTRACT FROM AUTHOR]