The article by Art Owen introduces the empirical likelihood ratio function and its application in constructing confidence regions for vector-valued statistical functionals. The key result is a nonparametric version of Wilks' theorem, which provides a multivariate generalization of Owen's earlier work. The Cornish-Fisher expansion demonstrates that empirical likelihood intervals for a one-dimensional mean are less affected by skewness compared to those based on Student's t statistic. Owen presents an effective method for computing empirical profile likelihoods for the mean of a vector random variable, reducing the problem to an unconstrained minimization of a convex function on a low-dimensional domain. Algorithms are available to find the unique global minimum with superlinear convergence. This method also yields 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, as well as statistics with linear estimating equations. An algorithm is provided for computing profile empirical likelihoods for these statistics. The article includes an example using data from Larsen and Marx (1986) to illustrate the empirical likelihood contours for the mean. It also discusses related literature, including the Bayesian bootstrap and the nonparametric tilting bootstrap, and compares empirical likelihood to other methods like Johnson's t and Student's t. The proof of the main theorem involves showing that the limit law of the empirical likelihood ratio is asymptotically chi-square distributed, with a rate of convergence of \(O(n^{-1/2})\). The article extends these results to other statistics, including nonlinear functionals and \(M\)-estimates, and provides examples of empirical likelihood inference for the variance, product moment correlation, and regression coefficients.The article by Art Owen introduces the empirical likelihood ratio function and its application in constructing confidence regions for vector-valued statistical functionals. The key result is a nonparametric version of Wilks' theorem, which provides a multivariate generalization of Owen's earlier work. The Cornish-Fisher expansion demonstrates that empirical likelihood intervals for a one-dimensional mean are less affected by skewness compared to those based on Student's t statistic. Owen presents an effective method for computing empirical profile likelihoods for the mean of a vector random variable, reducing the problem to an unconstrained minimization of a convex function on a low-dimensional domain. Algorithms are available to find the unique global minimum with superlinear convergence. This method also yields 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, as well as statistics with linear estimating equations. An algorithm is provided for computing profile empirical likelihoods for these statistics. The article includes an example using data from Larsen and Marx (1986) to illustrate the empirical likelihood contours for the mean. It also discusses related literature, including the Bayesian bootstrap and the nonparametric tilting bootstrap, and compares empirical likelihood to other methods like Johnson's t and Student's t. The proof of the main theorem involves showing that the limit law of the empirical likelihood ratio is asymptotically chi-square distributed, with a rate of convergence of \(O(n^{-1/2})\). The article extends these results to other statistics, including nonlinear functionals and \(M\)-estimates, and provides examples of empirical likelihood inference for the variance, product moment correlation, and regression coefficients.