© 1989 by Biometrika Trust
On adjustments based on the signed root of the empirical likelihood ratio statistic
Department of Statistics, Stanford University Stanford, California 94305, U.S.A.
The standard multivariate normal approximation to the distribution of the signed root of the empirical likelihood ratio statistic is considered in cases where inference is required for a smooth function of the mean of the distribution from which the sample is drawn. The error in this approximation is of order O(n–
) where n is the sample size, and it is shown that the error can be reduced to order O(n–1) by using a mean adjustment. When a scalar parameter is of interest, use of the mean adjustment produces confidence intervals whose endpoints each have coverage error of order O(n–1). This is in contrast to use of the unadjusted signed root, or equivalently, use of the chi-squared approximation for the empirical likelihood ratio statistic, which produces one-sided limits having coverage error of order O(n–). For a vector parameter of interest, a procedure asymptotically equivalent to the mean adjustment is developed that avoids explicit calculation of the signed square root. It is argued for the scalar parameter case that the coverage error of one-sided limits can be further reduced to O(n–3/2 by use of both mean and variance adjustments. The variance adjustment is given in the case when a scalar mean is of interest. The coverage accuracy of confidence limits obtained by the various methods is illustrated by simulation.
Key Words: Bartlett adjustment • Chi-squared approximation • Empirical likelihood ratio statistic • Nonparametric confidence region • Signed root empirical likelihood ratio statistic