In actuarial and insurance literatures, several researchers suggested generalized linear regression models (GLM) for modeling claim costs as a function of risk factors. The modeling of claim costs involving both zero and positive claims experience has been carried out by fitting the claim costs collectively using Tweedie model. However, the probability of zero claims in Tweedie model is not allowed to be fitted explicitly as a function of explanatory variables. The purpose of this article is to propose the application of Zero Adjusted Gamma (ZAGA) and Zero Adjusted Inverse Gaussian (ZAIG) regression models for modeling both zero and positive claim costs data. The models are fitted to the Malaysian motor insurance claims experiences which are divided into three types namely Third Party Bodily Injury (TPBI), Own Damage (OD) and Third Party Property Damage (TPPD). The fitted models show that both claim probability and claim cost are affected by either the same or different explanatory variables. The fitted models also allow the relative risk of each rating factor to be compared and the low or high risk vehicles to be identified, not only for the claim cost but also for the claim probability. The AIC and BIC indicate that ZAIG regression is the best model for modeling both positive and zero claim costs for all claim types