Replacement of Biased Estimators with Unbiased Ones in the Case of Student’s t-Distribution and Geary’s Kurtosis

Abstract

The use of biased estimators can be found in some historically and up to now important tools in statistical data analysis. In this paper their replacement with unbiased estimators at least in the case of the estimator of the population standard deviation for normal distributions is proposed. By removing the incoherence from the Student’s t-distribution caused by the biased estimator, a corrected t-distribution may be defined. Although the quantitative results in most data analysis applications are identical for both the original and corrected tdistributions, the use of this last t-distribution is suggested because of its theoretical consistency. Moreover, the frequent qualitative discussion of the t-distribution has come under much criticism, because it concerns artefacts of the biased estimator. In the case of Geary’s kurtosis the same correction results (2/π)1/2 unbiased estimation of kurtosis for normally distributed data that is independent of the size of the sample. It is believed that by removing the sample-size-dependent biased feature, the applicability domain can be expanded to include small sample sizes for some normality tests.

Keywords

• unbiased estimator
• normal distribution
• Anscombe-Glynn test
• Jarque-Bera test
• Bonett-Seier test