On the efficiencies of some common quick estimators

The quick estimators of location and scale have broad applications and are widely used. For a variety of symmetric populations we obtain the quantiles and the weights for which the asymptotic variances of the quick estimators are minimum. These optimal quick estimators are then used to obtain the asymptotic relative efficiencies of the commonly used estimators such as trimean, gastwirth, median, midrange, and interquartile range with respect to the optimal quick estimators in order to determine a choice among them and to check whether they are unacceptably poor. In the process it is seen that the interquartile range is the optimal quick estimator of scale for Cauchy populations; but the interdecile range is in general preferable. Also the optimal estimator of the location for the logistic distribution puts weights 0.3 on each of the two quartiles and 0.4 on the median. It is shown that for the symmetric distributions, such as the beta and Tukey-lambda with lambda > 0, which have finite support and short tails, i.e. the tail exponents (Parzen, 1979) satisfy gamma < 1, the midrange and the range are the optimal quick estimators of location and scale respectively if gamma < 1/2. The class of such distributions include the distributions with high discontinuous tails, e.g. Tukey-lambda with lambda > 1, as well as some distributions with p.d.f.'s going to zero at the ends of the finite support, such as Tukey-lambda with 1/2 < lambda < 1. As a byproduct an interesting tail correspondence between beta and Tukey-lambda distributions is seen.