Certain General Properties of Unbiased Estimates of Location and Scale Parameters Based on Ordered Observations

Abstract

Some upper bounds are derived for the variances of least squares estimators based on a subset of the ordered observations in a random sample (which may, for simplicity, be called best estimates or ordered least squares estimates) of (i) location, (ii) scale and (iii) both location and scale parameters of a distribution. The estimates achieving these upper bounds could, in some instances, be conservatively used in place of ordered least squares estimates. These results constitute generalizations of those due to Lloyd [6] and Downton [2], [3]. We also obtain Cramer-Rao type of lower bounds for the variance of a linear combination of censored observations (called a systematic statistic) which are valid even in the nonregular case. Finally, there is a discussion about an optimum choice of the function which can be used for generating lower bounds for variances of any estimators.