Skewness and kurtosis unbiased by Gaussian uncertainties

Abstract

Noise is an unavoidable part of most measurements which can hinder a correct interpretation of the data. Uncertainties propagate in the data analysis and can lead to biased results even in basic descriptive statistics such as the central moments and cumulants. Expressions of noise-unbiased estimates of central moments and cumulants up to the fourth order are presented under the assumption of independent Gaussian uncertainties, for weighted and unweighted statistics. These results are expected to be relevant for applications of the skewness and kurtosis estimators such as outlier detections, normality tests and in automated classification procedures. The comparison of estimators corrected and not corrected for noise biases is illustrated with simulations as a function of signal-to-noise ratio, employing different sample sizes and weighting schemes.