## The Annals of Statistics

### Empirical Likelihood Ratio Confidence Regions

Art Owen

#### Abstract

An empirical likelihood ratio function is defined and used to obtain confidence regions for vector valued statistical functionals. The result is a nonparametric version of Wilks' theorem and a multivariate generalization of work by Owen. Cornish-Fisher expansions show that the empirical likelihood intervals for a one dimensional mean are less adversely affected by skewness than are those based on Student's $t$ statistic. An effective method is presented for computing empirical profile likelihoods for the mean of a vector random variable. The method is a reduction by convex duality to an unconstrained minimization of a convex function on a low dimensional domain. Algorithms exist for finding the unique global minimum at a superlinear rate of convergence. A byproduct is a noncombinatorial algorithm for determining whether a given point lies within the convex hull of a finite set of points. The multivariate empirical likelihood regions are justified for functions of several means, such as variances, correlations and regression parameters and for statistics with linear estimating equations. An algorithm is given for computing profile empirical likelihoods for these statistics.

#### Article information

Source
Ann. Statist. Volume 18, Number 1 (1990), 90-120.

Dates
First available in Project Euclid: 12 April 2007

http://projecteuclid.org/euclid.aos/1176347494

JSTOR

Digital Object Identifier
doi:10.1214/aos/1176347494

Mathematical Reviews number (MathSciNet)
MR1041387

Zentralblatt MATH identifier
0712.62040

Subjects
Primary: 62E20: Asymptotic distribution theory

#### Citation

Owen, Art. Empirical Likelihood Ratio Confidence Regions. Ann. Statist. 18 (1990), no. 1, 90--120. doi:10.1214/aos/1176347494. http://projecteuclid.org/euclid.aos/1176347494.