The Zero-adjusted Inverse Gaussian distribution as a model for insurance claims.

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Inflated beta distributions (2007, unknown): “, Var(y) = αV1 + (1 − α)V2 + α(1 − α)(γ − μ)(2), (7)”
Bayesian Structured Additive Distributional Regression (2013, background): “In many applications, e.g. insurances [Klein et al., 2013a, Heller et al., 2006] or weather forecasts [Gneiting and Ranjan, 2011], distributions with point masses at zero are of great interest.”
Bayesian Generalized Additive Models for Location, Scale, and Shape for Zero-Inflated and Overdispersed Count Data (2015, methodology): “A detailed analysis based on both count data regression for claim frequencies and zero adjusted models (as introduced in Heller, Stasinopoulos, and Rigby 2006) for claim sizes in the framework of GAMLSS is provided in Klein et al. (2014).”
Nonlife Ratemaking and Risk Management with Bayesian Additive Models for Location, Scale and Shape (2013, methodology): “Zero-adjusted Gamma and Inverse-Gaussian models have been proposed by Heller et al. (2006), Bortoluzzo et al. (2011) and Resti et al. (2013) but their analysis only allowed for linear effects of the covariates, too.”