Linear moments study under ranked set sampling

Abstract

A ranked set sample consists of independently distributed order statistics and can occur naturally in many experimental settings. The weighted least squares method is used to find linear estimators of unknown parameters from location-scale family of distributions which required the full matrix of all variances and covariances of order statistics of the sample size which is difficult to obtain in large samples, finding the inverse of this matrix and the estimators can be computed numerically only for small sample sizes. Also, the weighted least squares can not be used when we have distribution which has more than two parameters, for example, generalized Pareto distribution. In this article, we are looking for method in the class of linear estimation which can be applied for any distribution under ranked set sample regardless of the number of the parameters and easy to use. The linear moment-L-moments- method does not require the full matrix of order statistics and easy to use. Also, we derive unbiased estimators of population L-moments using sample linear moments based on  independent ranked set sample. We obtain distribution-free estimate for the sample mean from any distribution under ranked set sample in terms of sample variance and sample L-moments. We illustrate our method on the generalized Pareto distribution.