Skewness and kurtosis unbiased by Gaussian uncertainties

TL;DR

This paper derives unbiased estimators for skewness and kurtosis in data affected by Gaussian noise, improving the accuracy of statistical analysis in noisy measurements.

Contribution

It provides explicit formulas for noise-unbiased estimates of third and fourth moments, applicable to weighted and unweighted data, under Gaussian noise assumptions.

Findings

Unbiased estimators reduce bias in skewness and kurtosis measurements.

Simulations show improved accuracy at various signal-to-noise ratios.

Corrected estimators outperform traditional biased ones in noisy conditions.

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.