SEMI-EMPIRICAL LIKELIHOOD RATIO CONFIDENCE INTERVALS FOR THE DIFFERENCE OF TWO SAMPLE MEANS

JING QIN

Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

(Received February 18, 1993; revised June 2, 1993)

Abstract. We all know that we can use the likelihood ratio statistic to test hypotheses and construct confidence intervals in full parametric models. Recently, Owen (1988, Biometrika, 75, 237–249; 1990, Ann. Statist., 18, 90–120) has introduced the empirical likelihood method in nonparametric models. In this paper, we combine these two likelihoods together and use the likelihood ratio to construct confidence intervals in a semiparametric problem, in which one model is parametric, and the other is nonparametric. A version of Wilks’s theorem is developed.

Key words and phrases: Empirical likelihood, hypotheses tests, semi-empirical likelihood, Wilks’s theorem.