This article extends uniform exposure credibility theory by making quadratic adjustments that take into account the squared values of past observations. This approach amounts to introducing non-linearities 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 non-parametric approach where premiums can only be estimated from data and no type of distribution is postulated. Finally, we examine the semi-parametric approach where the conditional distribution is Poisson but the unconditional distribution is unknown. For all of these approaches, the mean square error is, by construction, smaller in the q-credibility framework than in the standard framework.