Estimating a population median with a small sample

Abstract

Various estimators of the population median were compared for small samples (N ⩽ 50). The competitors included the sample median, Harrell-Davis (HD) estimate, six new estimates (Circle, DblExp, Exp+, Exp-, AltExp, Fibon) that require no à priori information about the population, and one estimate (AllExp) that requires information on the general form of the distribution The two benchmark criteria were the mean square error (MSE) and the rank based error (RBE). When there is no information available on the population, (a) AltExp was the best estimator for N ⩽ 30 for the MSE criterion, (b) the estimate based on the Fibonacci sequence was the best for N ⩽ 30 for the RBE criterion, and (c) the Harrell-Davis estimator was superior for N > 30. There was no support for using the sample median as an estimate of the population median.