COMBINED AND LEAST SQUARES EMPIRICAL LIKELIHOOD

BRUCE M. BROWN^{1} AND SONG XI CHEN^{2}

1Department of Mathematics, University of Tasmania, GPO Box 252L, Hobart, TAS 7001, Australia 2School of Statistical Science, La Trobe University, VIC 3083, Australia

(Received February 13, 1997; revised October 30, 1997)

Abstract. In conventional empirical likelihood, there is exactly one structural constraint for every parameter. In some circumstances, additional constraints are imposed to reflect additional and sought-after features of statistical analysis. Such an augmented scheme uses the implicit power of empirical likelihood to produce very natural adaptive statistical methods, free of arbitrary tuning parameter choices, and does have good asymptotic properties. The price to be paid for such good properties is in extra computational difficulty. To overcome the computational difficulty, we propose a ‘least-squares’ version of the empirical likelihood. The method is illustrated by application to the case of combined empirical likelihood for the mean and the median in one sample location inference.

Key words and phrases: Empirical likelihood, least squares empirical likelihood, maximum likelihood estimate, mean, median.