Generalized fiducial methods for testing the homogeneity of a three-sample problem with a mixture structure.

Area of Science:

• Statistics
• Statistical modeling
• Hypothesis testing

Background:

• The likelihood ratio (LR) test was recently introduced for homogeneity testing in three-sample mixture models.
• The LR test's null limiting distribution presents challenges for p-value determination due to its complex supremum form.
• The LR test exhibits suboptimal type-I error control, particularly with small to moderate sample sizes.

Purpose of the Study:

• To propose novel statistical methods for testing homogeneity in three-sample mixture models.
• To address the limitations of the existing likelihood ratio (LR) test, specifically its p-value calculation and type-I error control.

Main Methods:

• Development of seven generalized fiducial methods for homogeneity testing.
• Utilizing simulation studies to evaluate the performance of the proposed methods.
• Comparison of the proposed methods against the likelihood ratio (LR) test.

Main Results:

• The proposed generalized fiducial methods demonstrate superior control of type-I errors compared to the LR test, especially under small to moderate sample sizes.
• The new methods achieve comparable statistical power to the LR test in most scenarios.
• The effectiveness of the proposed methods is illustrated using a real-world halibut dataset.

Conclusions:

• Generalized fiducial methods provide a robust alternative for homogeneity testing in three-sample mixture models.
• These methods offer significant advantages in type-I error rate control for limited sample data.
• The proposed techniques are practical and applicable, as shown by the analysis of the halibut data.