A Space-Efficient Recursive Procedure for Estimating a Quantile of an Unknown Distribution

Abstract

Consider the problem of computing an estimate of a percentile or quantile of an unknown population based on a random sample of n observations. By viewing this problem as a problem in stochastic approximation, we obtain an estimator that requires only a small amount of direct access storage space that does not increase with the sample size. We show that a modified version of the simple stochastic approximation estimator has the same large-sample behavior as the sample quantile, which has the smallest asymptotic variance among all reasonable estimators. The modified procedure also yields an estimate of the asymptotic variance of the estimator. Some simulation results are presented to show that the proposed estimator performs well in samples of moderate size.