Least-Squares Estimates Using Ordered Observations

Abstract

The purpose of this paper is to compare for various two-parameter distributions, of the form $f\{(x - \mu)/\sigma\}/\sigma,$ the estimates of the parameters obtained by applying the method of least squares to the observations, after these have been arranged in order of magnitude. Estimates obtained by this process we shall call "ordered least-squares estimates." Such estimates are unbiased and have minimal variance among all unbiased estimates which are linear in the ordered observations. This estimation process has been previously discussed by Godwin [1] and [2] and Lloyd [3]. In the present paper, ordered estimates are obtained explicitly for a class of two-parameter distributions having the above form. This class contains the rectangular and the right triangular distributions as special cases. It also reduces to the exponential distribution as a limiting case. Other special cases of this class of distributions have also been previously discussed by Craig [4]. Further, a general property of ordered least-squares estimates of the parameter $\lambda$ in distributions of the type $f(x/\lambda)/\lambda$ is discussed. As a result it is shown that the ordered least-squares estimate of the scale parameter in the Pearson Type III distribution is identical with the maximum likelihood estimate.