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COMBINED AND LEAST SQUARES EMPIRICAL LIKELIHOOD

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

^{1} *Department of Mathematics, University of Tasmania,*

GPO Box 252L, Hobart, TAS 7001, Australia

^{2} *School 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.

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