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--- abstract: 'In this paper, we propose two important extensions to cluster-weighted models (CWMs). First, we extend CWMs to have generalized cluster-weighted models (GCWMs) by allowing modeling of non-Gaussian distribution of the continuous covariates, as they frequently occur in insurance practice. Secondly, we introduce a zero-inflated extension of GCWM (ZI-GCWM) for modeling insurance claims data with excess zeros coming from heterogenous sources. Additionally, we give two expectation-optimization (EM) algorithms for parameter estimation given the proposed models. An appropriate simulation study shows that, for various settings and in contrast to the existing mixture-based approaches, both extended models perform well. Finally, a real data set based on French auto-mobile policies is used to illustrate the application of the proposed extensions.' author: - 'Nikola Počuča$^*$, Petar Jevti'' c$^{**}$, Paul D. McNicholas$^*$, and Tatjana Miljkovic$^{\dagger}$' date: | $^*$Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada.\ $^{**}$Department of Mathematics and Statistics, Arizona State University, Phoenix, AZ, U.S.\ $^{\dagger}$Department of Statistics, Miami University, Oxford, OH, U.S. title: 'Modeling Frequency and Severity of Claims with the Zero-Inflated Generalized Cluster-Weighted Models' --- <span style="font-variant:small-caps;">Key Words:</span> GCWM, CWM, clustering, automobile claims.\ <span style="font-variant:small-caps;">KEL Classification:</span> C02, C40, C60.\ Introduction {#sec:introduction} ============ A significant number of clustering methods have been proposed for sub-grouping the data in the area of computer science, biology, social science, statistics, marketing, etc. [@Ingrassia+Punzo+Vittadini+Minotti:2015] proposed a cluster-weighted model (CWM) framework as a flexible family of mixture models for fitting the joint distribution of a random vector composed of a response variable and a set of mixed-type covariates with the assumption that continuous covariates come from Gaussian distribution. CWMs with Gaussian assumptions have been proposed by [@Gershenfeld:1997], [@Gershenfeld:Schoner+Metois:1999], and [@Gershenfeld:1999] in a context of media technology. Some extensions of this class of models have been considered by [@Punzo+Ingrassia:2015], [@Ingrassia+Minotti+Punzo:2014], [@Ingrassia+Minotti+Vittadini:2012], [@subedi13; @subedi15], and [@punzo17]. These clustering methods are somewhat lacking for modelling insurance data, e.g., high excess zeros for claim count, heavy-tail loss distribution, deductible, or limits. Sub-grouping of insurance policies based on risk classification is a standard practice in insurance. The heterogenous nature of insurance data allows for explorations of many different techniques for sub-grouping risk. As a result, there is a growing number of papers in the area of mixture modeling of univariate and multivariate insurance data to account for heterogeneity of risk. [@Lee+Lin:2010], [@Verbelen+Gong+Antonio+Badescu+Lin:2015], and [@Miljkovic+Grun:2016] proposed mixture models for univariate loss data, and mixture modeling of univariate insurance data has been extended to the multivariate context. A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking was studied by [@Bermudez+Karlis:2012]. A Poisson mixture model for count data was considered by [@Brown+Buckley:2015] with application in managing a Group Life insurance portfolio. Recently, [@risks_miljkovic] reviewed two complementary mixture-based clustering approaches (CWMs and mixture-based clustering for an ordered stereotype model) for modeling unobserved heterogeneity in an automobile insurance portfolio, depending on the data structure under consideration. In this paper, we extend the CWM family proposed by [@Ingrassia+Punzo+Vittadini+Minotti:2015] to allow for modeling of non-Gaussian continuous covariates and a zero-inflated Poisson (ZIP) claims