The Ubiquitous Role of f'/f in Efficient Estimation of Location

There has been considerable recent discussion as to what location estimator one should use.Traditionally, the sample mean has been used, often in conjunction with formal or informal outlier rejection rules.However, there may be considerable loss in efficiency if the mean is used on data for which it is not appropriate.This expository article seeks to provide insight into the estimation problem.It focuses on the all pervasive importance of f'/f in efficient estimation of location, where f is the density function of the distribution generating the data and f' is its derivative.Connections are discussed among the three major classes of location estimators dominant in the robustness literature: M estimators (maximum likelihood-like), R (rank) estimators, and L estimators (linear combinations of order statistics).In all three classes f'/f is seen to be the key to efficient estimation.A heuristic explanation is given as to why f'/f is an intuitively reasonable quantity on which to base estimation.The asymptotic relative efficiency of all three classes of estimators is seen to be the square of a correlation coefficient in f'/f.