Improved statistics for contrasting means of two samples under non-normality

Area of Science:

• Statistics
• Statistical Inference
• Hypothesis Testing

Background:

• Standard statistical tests often assume normal distributions, which may not hold in practice.
• Non-normality can lead to inaccurate conclusions when comparing two independent populations.
• Existing methods for non-normal data may lack sufficient precision.

Purpose of the Study:

• To develop improved asymptotic expansions for the two-sample t-statistic and Welch statistic.
• To enhance the precision of statistical inference under non-normality.
• To provide practical guidelines for applying these improved statistics.

Main Methods:

• Deriving null distributions using asymptotic expansions up to n(-1) for the standard normal variable.
• Employing monotone transformations to mitigate higher-order cumulant effects.
• Conducting numerical simulations to evaluate the performance of the proposed statistics.

Main Results:

• The new statistics achieve a higher level of precision, specifically o(n(-1)).
• Asymptotic expansions are expressed in terms of standard normal distribution and density functions.
• Numerical studies confirm the improved accuracy and performance of the enhanced statistics.

Conclusions:

• The proposed methods offer a more precise approach to testing the equality of means for independent populations with non-normal distributions.
• The enhanced statistics provide a valuable tool for improving the reliability of statistical inference.
• Practical recommendations are provided for researchers and practitioners using these advanced statistical techniques.