Estimating the parameters of the normal, exponential and gamma distributions using median and extreme ranked set samples

Abstract

In this paper, we propose maximum likelihood estimators (mle’s) as well as linear un-biased estimators (lue’s) of the parameters of the normal, exponential and gamma distri-butions in the light of the location-scale family of distributions - i.e. distributions with cumulative distribution functions of the form F ((x – µ)/?), using median ranked set sam-pling (MRSS) and extreme ranked set sampling (ERSS). MRSS and ERSS are modifica-tions of ranked set sampling (RSS), which are more practicable and less prone to prob-lems resulting from erroneous ranking. The mle’s of the normal mean and the scale pa-rameters of the exponential and gamma distributions under MRSS are shown to dominate all other estimators, while the mle of the normal standard deviation under ERSS is the most efficient. A similar trend is observed in the lue’s. A modification of ERSS namely partial extreme ranked set sampling (PERSS) is proposed for odd set sizes to generate even-sized samples. The lue of the normal standard deviation under this modification is shown to be the most efficient of all the lue’s of the same parameter. Among the lue’s considered, the PERSS lue’s are the most efficient when the sample size per cycle is two.