On non-parametric and generalized tests for the two-sample problem with location and scale change alternatives

Area of Science:

• Statistics
• Statistical Inference
• Hypothesis Testing

Background:

• Traditional statistical tests often assume data normality, limiting their applicability.
• The two-sample problem, particularly with general alternatives, requires robust testing methods.
• Existing generalized tests may lack flexibility or desirable properties like level robustness.

Purpose of the Study:

• To establish a direct relationship between O'Brien's generalized tests and Lepage's non-parametric tests.
• To develop a broader, more adaptable class of O'Brien-type procedures.
• To evaluate the performance of these novel procedures regarding empirical size and power.

Main Methods:

• Establishing theoretical links between generalized t-tests, rank sum tests, and non-parametric Lepage tests.
• Developing a new family of O'Brien-type procedures.
• Conducting simulations to compute empirical sizes and powers across various data models.

Main Results:

• Generalized O'Brien tests are shown to be directly related to Lepage's non-parametric tests.
• A flexible class of O'Brien-type procedures with inherited level robustness was developed.
• Empirical evaluations demonstrated the validity and good power of non-parametric and O'Brien-type tests.

Conclusions:

• The proposed O'Brien-type tests are valid and powerful alternatives for the two-sample problem.
• These tests are superior to traditional t-tests, especially for skewed or long-tailed distributions.
• The enhanced procedures offer improved statistical reliability and flexibility in hypothesis testing.