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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]

Summary: Usually in latent class (LC) analysis, external predictors are taken to be cluster conditional probability predictors (LC models with external predictors), and/or score conditional probability predictors (LC regression models). In such cases, their distribution is not of interest. Class‐specific distribution is of interest in the distal outcome model, when the distribution of the external variables is assumed to depend on LC membership. In this paper, we consider a more general formulation, that embeds both the LC regression and the distal outcome models, as is typically done in cluster‐weighted modelling. This allows us to investigate (1) whether the distribution of the external variables differs across classes, (2) whether there are significant direct effects of the external variables on the indicators, by modelling jointly the relationship between the external and the latent variables. We show the advantages of the proposed modelling approach through a set of artificial examples, an extensive simulation study and an empirical application about psychological contracts among employees and employers in Belgium and the Netherlands. [ABSTRACT FROM AUTHOR]

Compositional data have peculiar characteristics that pose significant challenges to traditional statistical methods and models. Within this framework, we use a convenient mode parametrized Dirichlet distribution across multiple fields of statistics. In particular, we propose finite mixtures of unimodal Dirichlet (UD) distributions for model-based clustering and classification. Then, we introduce the contaminated UD (CUD) distribution, a heavy-tailed generalization of the UD distribution that allows for a more flexible tail behavior in the presence of atypical observations. Thirdly, we propose finite mixtures of CUD distributions to jointly account for the presence of clusters and atypical points in the data. Parameter estimation is carried out by directly maximizing the maximum likelihood or by using an expectation-maximization (EM) algorithm. Two analyses are conducted on simulated data to illustrate the effects of atypical observations on parameter estimation and data classification, and how our proposals address both aspects. Furthermore, two real datasets are investigated and the results obtained via our models are discussed. [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]

This article shows how multivariate elliptically contoured (EC) distributions, parameterized according to the mean vector and covariance matrix, can be built from univariate standard symmetric distributions. The obtained distributions are referred to as moment-parameterized EC (MEC) herein. As a further novelty, the article shows how to polynomially reshape MEC distributions and obtain distributions, called leptokurtic MEC (LMEC), having probability density functions characterized by a further parameter expressing their excess kurtosis with respect to the parent MEC distributions. Two estimation methods are discussed: the method of moments and the maximum likelihood. For illustrative purposes, normal, Laplace, and logistic univariate densities are considered to build MEC and LMEC models. An application to financial returns of a set of European stock indexes is finally presented. [ABSTRACT FROM AUTHOR]

Summary: The expectation–maximization (EM) algorithm is a familiar tool for computing the maximum likelihood estimate of the parameters in hidden Markov and semi‐Markov models. This paper carries out a detailed study on the influence that the initial values of the parameters impose on the results produced by the algorithm. We compare random starts and partitional and model‐based strategies for choosing the initial values for the EM algorithm in the case of multivariate Gaussian emission distributions (EDs) and assess the performance of each strategy with different assessment criteria. Several data generation settings are considered with varying number of latent states, of variables as well as of the level of fuzziness in the data, and discussion on how each factor influences the obtained results is provided. Simulation results show that different initialization strategies may lead to different log‐likelihood values and, accordingly, to different estimated partitions. A clear indication of which strategies should be preferred is given. We further include two real‐data examples, widely analysed in the hidden semi‐Markov model literature. [ABSTRACT FROM AUTHOR]

The multivariate contaminated normal (MCN) distribution represents a simple heavy-tailed generalization of the multivariate normal (MN) distribution to model elliptical contoured scatters in the presence of mild outliers (also referred to as 'bad' points herein) and automatically detect bad points. The price of these advantages is two additional parameters: proportion of good observations and degree of contamination. However, in a multivariate setting, only one proportion of good observations and only one degree of contamination may be limiting. To overcome this limitation, we propose a multiple scaled contaminated normal (MSCN) distribution. Among its parameters, we have an orthogonal matrix Γ. In the space spanned by the vectors (principal components) of Γ, there is a proportion of good observations and a degree of contamination for each component. Moreover, each observation has a posterior probability of being good with respect to each principal component. Thanks to this probability, the method provides directional robust estimates of the parameters of the nested MN and automatic directional detection of bad points. The term 'directional' is added to specify that the method works separately for each principal component. Mixtures of MSCN distributions are also proposed, and an expectation-maximization algorithm is used for parameter estimation. Real and simulated data are considered to show the usefulness of our mixture with respect to well-established mixtures of symmetric distributions with heavy tails. [ABSTRACT FROM AUTHOR]

We revisit the generalized hyperbolic (GH) distribution and its nested models. These include widely used parametric choices like the multivariate normal, skew‐t$$ t $$, Laplace, and several others. We also introduce the multiple‐choice LASSO, a novel penalized method for choosing among alternative constraints on the same parameter. A hierarchical multiple‐choice Least Absolute Shrinkage and Selection Operator (LASSO) penalized likelihood is optimized to perform simultaneous model selection and inference within the GH family. We illustrate our approach through a simulation study and a real data example. The methodology proposed in this paper has been implemented in R functions which are available as supplementary material. [ABSTRACT FROM AUTHOR]

Two families of matrix-variate hidden Markov regression models (MV-HMRMs) are here introduced. The distinction between them relies on the role of the covariates, which can be treated as fixed or random. Parsimony is achieved by using the eigen-decomposition of the components’ covariance matrices. This generates a different number of parsimonious models between the two families: 98 MV-HMRMs with fixed covariates and 9604 MV-HMRMs with random covariates. ECM algorithms are discussed for parameter estimation and, because of the high number of parsimonious models, convenient initialization, and fitting strategies are proposed. Our models are first applied to simulated data, and then to a four-way real dataset concerning the relationship between unemployment and labor force participation in the Italian provinces. [ABSTRACT FROM AUTHOR]

We propose the family of dimension-wise scaled normal mixtures (DSNMs) to model the joint distribution of a d -variate random variable with real-valued components. Each member of the family generalizes the multivariate normal (MN) distribution in two directions. Firstly, the DSNM has a more general type of symmetry with respect to the elliptical symmetry of the MN distribution. Secondly, the univariate marginals have similar heavy-tailed normal scale mixture distributions with (possibly) different tailedness parameters; as a consequence of practical interest, the DSNM allows for a different excess kurtosis on each dimension. These peculiarities are obtained in an MN scale mixture framework by introducing a d -variate mixing random variable with independent and similar components acting separately for each dimension. We examine several properties of DSNMs, such as the joint density function, hierarchical and stochastic representations, relations with other families of symmetric distributions, type of symmetry, marginal and conditional distributions, no correlation implying independence, moment generating function, and moments of practical interest. For illustrative purposes, we describe two members of the DSNM family obtained in the case of components of the mixing random vector being either uniform or shifted exponential; these are examples of mixing distributions that guarantee a closed-form expression for the joint density of the DSNM. For the two DSNMs analyzed in detail, we introduce parsimony by allowing the d tailedness parameters to be tied across dimensions, and describe algorithms, based on the expectation–maximization (EM) principle, to estimate the parameters by maximum likelihood. We use real data from the financial and biometrical fields to appreciate the advantages of our DSNMs over other symmetric heavy-tailed distributions available in the literature. [ABSTRACT FROM AUTHOR]

In generalized linear models (GLMs), measures of lack of fit are typically defined as the deviance between two nested models, and a deviance-based R2 is commonly used to evaluate the fit. In this paper, we extend deviance measures to mixtures of GLMs, whose parameters are estimated by maximum likelihood (ML) via the EM algorithm. Such measures are defined both locally, i.e., at cluster-level, and globally, i.e., with reference to the whole sample. At the cluster-level, we propose a normalized two-term decomposition of the local deviance into explained, and unexplained local deviances. At the sample-level, we introduce an additive normalized decomposition of the total deviance into three terms, where each evaluates a different aspect of the fitted model: (1) the cluster separation on the dependent variable, (2) the proportion of the total deviance explained by the fitted model, and (3) the proportion of the total deviance which remains unexplained. We use both local and global decompositions to define, respectively, local and overall deviance R2 measures for mixtures of GLMs, which we illustrate—for Gaussian, Poisson and binomial responses—by means of a simulation study. The proposed fit measures are then used to assess, and interpret clusters of COVID-19 spread in Italy in two time points. [ABSTRACT FROM AUTHOR]

Analysis of matrix-variate data is becoming ever more prevalent in the literature, especially in the area of clustering and classification. Real data, including real matrix-variate data, are often contaminated by potential outlying observations. Their detection, as well as the development of models insensitive to their presence, is particularly important for this type of data because of the practical issues concerning their effective visualization. Herein, the matrix-variate contaminated normal distribution is discussed and then utilized in the mixture model paradigm for clustering. One key advantage of the proposed model is the ability to automatically detect potential outlying matrices by computing their a posteriori probability of being typical or atypical. Such detection is currently unavailable using existing matrix-variate methods. An expectation conditional maximization algorithm is used for parameter estimation, and both simulated and real data are used for illustration. Supplementary files for this article are available online. [ABSTRACT FROM AUTHOR]

Finite mixtures of regressions with fixed covariates are a commonly used model-based clustering methodology to deal with regression data. However, they assume assignment independence, i.e., the allocation of data points to the clusters is made independently of the distribution of the covariates. To take into account the latter aspect, finite mixtures of regressions with random covariates, also known as cluster-weighted models (CWMs), have been proposed in the univariate and multivariate literature. In this paper, the CWM is extended to matrix data, e.g., those data where a set of variables are simultaneously observed at different time points or locations. Specifically, the cluster-specific marginal distribution of the covariates and the cluster-specific conditional distribution of the responses given the covariates are assumed to be matrix normal. Maximum likelihood parameter estimates are derived using an expectation-conditional maximization algorithm. Parameter recovery, classification assessment, and the capability of the Bayesian information criterion to detect the underlying groups are investigated using simulated data. Finally, two real data applications concerning educational indicators and the Italian non-life insurance market are presented. [ABSTRACT FROM AUTHOR]

Cluster-weighted models (CWMs) extend finite mixtures of regressions (FMRs) in order to allow the distribution of covariates to contribute to the clustering process. In this article, we introduce 24 matrix-variate CWMs which are obtained by allowing both the responses and covariates in each cluster to be modelled by one of four existing skewed matrix-variate distributions or the matrix-variate normal distribution. Endowed with greater flexibility, our matrix-variate CWMs are able to handle this kind of data in a more suitable manner. As a by-product, the four skewed matrix-variate FMRs are also introduced. Maximum likelihood parameter estimates are derived using an expectation-conditional maximization algorithm. Parameter recovery, classification assessment, and the capability of the Bayesian information criterion to detect the underlying groups are investigated using simulated data. Lastly, our matrix-variate CWMs, along with the matrix-variate normal CWM and matrix-variate FMRs, are applied to two real datasets for illustrative purposes. [ABSTRACT FROM AUTHOR]

We propose a general approach to detect measurement non-invariance in latent Markov models for longitudinal data. We define different notions of differential item functioning in the context of panel data. We then present a model selection approach based on the Bayesian information criterion (BIC) to choose both the number of latent states and the measurement structure. We show the practical relevance by means of an extensive simulation study, and illustrate its use on two real–data examples from the social sciences. Our results indicate that BIC is able to select the correct measurement equivalence structure more than 95% of times. [ABSTRACT FROM AUTHOR]