Problematic standard errors and confidence intervals for skewness and kurtosis

Area of Science:

• Statistics
• Data Analysis
• Statistical Modeling

Background:

• Standard statistical software often provides standard errors for skewness and kurtosis.
• These standard errors typically assume a normal distribution, which is frequently not met in real-world data.
• Commonly used textbook heuristics for hypothesis testing based on these standard errors may be inappropriate.

Purpose of the Study:

• To evaluate the accuracy of traditional standard error estimates for skewness and kurtosis.
• To compare traditional methods with bootstrapping for calculating standard errors and confidence intervals.
• To recommend improved methods for statistical inference involving skewness and kurtosis.

Main Methods:

• Analysis of data distributions where traditional standard error calculations were compared to actual values.
• Application of the bootstrap method to estimate standard errors and confidence intervals for skewness and kurtosis.
• Utilizing freeware R for implementing bootstrap calculations.

Main Results:

• Traditional standard errors can significantly underestimate or overestimate true values, sometimes by a factor of 5.
• Bootstrap standard errors and confidence intervals demonstrate greater accuracy compared to traditional methods.
• While imperfect, bootstrapping offers a more robust approach to statistical inference for these moments.

Conclusions:

• The traditional method for calculating standard errors of skewness and kurtosis is unreliable due to its assumption of normality.
• Bootstrapping is recommended as a more accurate and robust method for estimating standard errors and confidence intervals for skewness and kurtosis.
• Freeware R implementations are available to facilitate the use of bootstrapping for these statistical analyses.