Estimation of Quantiles of Location-Scale Distributions Based on Two or Three Order Statistics

Abstract

Linear asymptotically unbiased estimators of ξ quantiles, x _ξ 0 < ξ < 1, of location-scale distributions are considered. These are based on two or three order statistics suitably selected in a neighborhood of the sample quantile X _(N), N = [nξ] + 1, where n is the sample size. The estimators are easy to calculate and are substantially more efficient than the nonparametric estimator _ξ = X _(N). The estimators are tabulated for selected values of ξ for the normal and extreme value (Gumbel) distributions. Also given are the asymptotic relative efficiencies of these estimators when compared with the maximum likelihood estimator of x _ξ based on all n observations.

KEY WORDS:

• Estimation
• Order statistics
• ξ quantile