This paper extends uniform-exposure credibility theory by making quadratic adjustments that take into account the squared values of past observations. This approach amounts to introducing nonlinearities in the framework, or to considering higher-order cross-moments in the computations. We first describe the full parametric approach and, for illustration, we examine the Poisson-gamma and Poisson-Pareto cases. Then, we look at the nonparametric approach, whereby premiums can be estimated only from data and no type of distribution is postulated. Finally, we examine the semiparametric approach, in which the conditional distribution is Poisson but the unconditional distribution is unknown. For all of these approaches, the mean squared error is, by construction, smaller in the q -credibility framework than in the standard framework. Address for Correspondence: lecourtois@em-lyon.com