Estimation of the scale parameter of a log-logistic distribution
In this work the ranked set sampling technique has been applied to estimate the scale parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> </InlineEquation>. Accordingly we have used median ranked set sampling to estimate <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> </InlineEquation>. We have further carried out the multistage median ranked set sampling to estimate <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$Z$$</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$(X, Z)$$</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$n$$</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> </InlineEquation> using the measurements made on those units in this ranked set sample. Copyright Springer-Verlag 2013
