Empirical likelihood ratio test for symmetry against type I bias with applications to competing risks

Abstract

A random variable X with cumulative distribution function F is said to have a symmetric distribution about θ if and only if X−θ and−X+θ are identically distributed. Different types of partial skewness and one-sided bias are obtained by looking at different types of orderings between the distributions of X−θ and−X+θ. For example, X, or equivalently F, is said to have type I bias about θ if X−θ is stochastically larger than−X+θ. In this paper, we assume that F is continuous, θ is known and develops an empirical likelihood ratio type test for testing for symmetry about θ against this type of alternative. This test is shown to be asymptotically distribution free and the results of a simulation study show that it outperforms in terms of power, a test developed for the same problem in Alfieri and El Barmi [(2005), ‘Nonparametric Estimation of a Distribution Function with Type I Bias with Applications to Competing Risks’, Journal of Nonparametric Statistics, 17, 319–333]. It turns out that the results developed here can be extended in a natural way to compare the sub-survival functions corresponding to two risks in a competing risks setting. We show how this can be done and illustrate our theoretical results with a real life example.

Keywords:

• convergence in distribution
• competing risks
• hypotheses testing
• symmetry
• type I bias

AMS 1991 Subject Classifications :

• 62P10
• 62G05
• 62E10

Acknowledgements

The authors are grateful to the editor and two referees whose comments have resulted in a much improved paper. Part of this work was carried out while Hammou El Barmi was visiting Faculte des Sciences et Techniques, Universite Moulay Ismail, Maroc. His work is supported by the City University of New York through PSC-CUNY.