Min-max approach for comparison of univariate normality tests

Area of Science:

• Statistics
• Statistical Hypothesis Testing

Background:

• Normality tests often yield ambiguous results because their power depends on unspecified alternative distributions.
• A single test's optimality against one alternative distribution does not guarantee performance against others.
• Existing invariant benchmarks, while robust, involve computationally intensive Neyman-Pearson tests against all alternatives.

Purpose of the Study:

• To propose a computationally efficient alternative approach for calculating an invariant benchmark for normality tests.
• To reduce the computational cost associated with estimating Neyman-Pearson tests against diverse alternative distributions.

Main Methods:

• Development of a novel min-max approach to approximate the benchmark calculation.
• Extensive simulation studies were conducted to evaluate various normality tests using the proposed methodology.
• Comparison of the proposed min-max method's performance against the established Neyman-Pearson benchmark.

Main Results:

• The proposed min-max approach significantly reduces computational cost compared to traditional methods.
• The min-max method yields results comparable to the Neyman-Pearson benchmark for evaluating normality tests.
• Selected normality tests demonstrated varying performance, as evaluated by the proposed simulation framework.

Conclusions:

• The min-max approach provides a computationally feasible and accurate method for benchmarking normality tests.
• This methodology offers a practical solution for researchers needing to assess normality test performance without prohibitive computational expense.
• The study validates the utility of the min-max approach in statistical hypothesis testing for normality.