This thesis examines the problem of accurately estimating the expected value and variance of the aggregate claims for each policyholder. Through an appropriate statistical model to estimate the pure premium, an insurer can find niche markets to operate competitively and profitably. To this end, the framework of generalized linear models (GLMs) for aggregate claims is extended to a structure of frequentist generalized additive models (GAMs) based on cubic penalized regression splines. The new structure could allow more flexible nonlinear and/or nonparametric trend terms for the marginal claim frequency and conditional claim severity models, as well as the Tweedie modelling. Then, this nonparametric approach is illustrated through simulation and applied to an automobile insurance dataset. The hypothesis tests' results, Akaike's Information Criterion (AIC) values and graphical diagnostics all show that the GAMs under both the independent and dependent settings give a better fit than the corresponding previously proposed parametric approach.