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Circular statistics and Rayleigh tests are important tools for analyzing cyclic events. However, current methods are not robust to significant measurement bias, especially incomplete or otherwise non-uniform sampling. One example is studying 24-cyclicity but having data not recorded uniformly over the full 24-hour cycle. Our objective is to present a robust method to estimate circular statistics and their statistical significance in the presence of incomplete or otherwise non-uniform sampling. Our method is to solve the underlying Fredholm Integral Equation for the more general problem, estimating probability distributions in the context of imperfect measurements, with our circular statistics in the presence of incomplete/non-uniform sampling being one special case. The method is based on linear parameterizations of the underlying distributions. We simulated the estimation error of our approach for several toy examples as well as for a real-world example: analyzing the 24-hour cyclicity of an electrographic biomarker of epileptic tissue controlled for states of vigilance. We also evaluated the accuracy of the Rayleigh test statistic versus the direct simulation of statistical significance. Our method shows a very low estimation error. In the real-world example, the corrected moments had a root mean square error of   $\bm {&lt; 0.007}$  . In contrast, the Rayleigh test statistic overestimated the statistical significance and was thus not reliable. The presented methods thus provide a robust solution to computing circular moments even with incomplete or otherwise non-uniform sampling. Since Rayleigh test statistics cannot be used in this circumstance, direct estimation of significance is the preferable option for estimating statistical significance.

Estimation of Circular Statistics in the Presence of Measurement Bias