Weighted skewness and kurtosis unbiased by sample size

Abstract

Central moments and cumulants are often employed to characterize the distribution of data. The skewness and kurtosis are particularly useful for the detection of outliers, the assessment of departures from normally distributed data, automated classification techniques and other applications. Robust definitions of higher order moments are more stable but might miss characteristic features of the data, as in the case of astronomical time series with rare events like stellar bursts or eclipses from binary systems. Weighting can help identify reliable measurements from uncertain or spurious outliers, so unbiased estimates of the weighted skewness and kurtosis moments and cumulants, corrected for sample-size biases, are provided under the assumption of independent data. The comparison of biased and unbiased weighted estimators is illustrated with simulations as a function of sample size, employing different data distributions and weighting schemes.