An Empirical Likelihood Ratio Test for Normality

Abstract

The empirical likelihood ratio (ELR) test for the problem of testing for normality is derived in this article. The sampling properties of the ELR test and four other commonly used tests are provided and analyzed using the Monte Carlo simulation technique. The power comparisons against a wide range of alternative distributions show that the ELR test is the most powerful of these tests in certain situations.

Keywords:

• Empirical likelihood
• Monte Carlo simulation
• Testing for normality

Mathematics Subject Classification:

• Primary 62F04
• Secondary 62G10

Acknowledgments

This article is based on one chapter of the first author's Ph.D. dissertation, completed in the Department of Economics, University of Victoria, in December 2003. Special thanks go to Don Ferguson, Ron Mittelhammer, Min Tsao, Graham Voss, and Julie Zhou for their many helpful suggestions and contributions.

Notes

Notes: m and n are the number of replications and the sample size. The true values of the parameters (μ, σ²)′ = (0, 1)′. The χ² tests may not be applicable with some small sample sizes.

Notes: The number of replications is 10,000. n is the sample size. The true values of the parameters (μ, σ²)′ = (0, 1)′. The χ² tests may not be applicable with some small sample sizes.

Notes: The number of replications is 10,000. n is the sample size. The true values of the parameters (μ, σ²)′ = (0, 1)′. The χ² tests may not be applicable with some small sample sizes.

Notes: The number of replications is 10,000. n is the sample size. The data is standardized to be x _i  = (y _i  − μ)/σ, for i = 1, 2,…, n. The true values of the parameters (μ, σ²)′ = (0, 1)′.

Notes: The number of replications is 10,000. n is the sample size. The true values of the parameters (μ, σ²)′ = (0, 1)′. ELR₄ and ELR₅ are the ELR test with four and five moment equations, respectively. The degrees of freedom of the ELR₅ test is 3. The alternative of student t ₍₅₎ is not applicable in this excise.