The assumptions of paleointensity experiments are violated in many natural and archaeological materials, leading to Arai plots which do not appear linear and yield inaccurate paleointensity estimates, leading to bias in the result. Recently, paleomagnetists have adopted sets of “selection criteria” that exclude specimens with non linear Arai plots from the analysis, but there is little consensus in the paleomagnetic community on which set to use. In this paper, we present a statistical method we call Bias Corrected Estimation of Paleointensity (BiCEP), which assumes that the paleointensity recorded by each specimen is biased away from a true answer by an amount that is dependent a single metric of nonlinearity (the curvature parameter $\vec{k}$) on the Arai plot. We can use this empirical relationship to estimate the recorded paleointensity for a specimen where $\vec{k}=0$, i.e., a perfectly straight line. We apply the BiCEP method to a collection of 30 sites for which the true value of the original field is well constrained. Our method returns accurate estimates of paleointensity, with a higher level of accuracy and precision than the strict CCRIT selection criteria, and with higher accuracy and similar precision to the modified PICRIT03 criteria. The BiCEP method has a significant advantage over using these selection criteria because it achieves these accurate results without excluding large numbers of specimens from the analysis.