Estimation of the scale parameter of a log-logistic distribution

Abstract

In this work the ranked set sampling technique has been applied to estimate the scale parameter \(\alpha \) of a log-logistic distribution under a situation where the units in a sample can be ordered by judgement method without any error. We have evaluated the Fisher information contained in the order statistics arising from this distribution and observed that median of a random sample contains the maximum information about the parameter \(\alpha \). Accordingly we have used median ranked set sampling to estimate \(\alpha \). We have further carried out the multistage median ranked set sampling to estimate \(\alpha \) with improved precision. Suppose it is not possible to rank the units in a sample according to judgement method without error but the units can be ordered based on an auxiliary variable \(Z\) such that \((X, Z)\) has a Morgenstern type bivariate log-logistic distribution (MTBLLD). In such a situation we have derived the Fisher information contained in the concomitant of rth order statistic of a random sample of size \(n\) from MTBLLD and identified those concomitants among others which possess largest amount of Fisher information and defined an unbalanced ranked set sampling utilizing those units in the sample and thereby proposed an estimator of \(\alpha \) using the measurements made on those units in this ranked set sample.