We develop a new algorithm for binary event detection in wireless sensor networks where the sensing process exhibits non-linear behavior causing clipping of the signals at random. We formulate the optimal event detection as a likelihood ratio test under a mixture model, which results in intractable marginal likelihood functions. To evaluate the intractable marginal likelihood functions, we develop a novel approximation via a non-parametric probability density estimator that is based on a series expansion of the Beta-Jacobi family of basis functions. The advantage of our algorithm is that it provides both high detection rate as well as very low computational complexity. In fact, our algorithm only requires the calculation of the first few moments of the model, which can be evaluated offline and is simple to implement in practice. Another feature of our model is that it is general enough to incorporate the probabilistic sensor non-linearities as well as transmission over random wireless channels. Simulation results demonstrate the benefits of using our framework, and that accounting for such practical features significantly improves the detection performance compared with the widely used Gaussian approximation method.