Order statistics of uniform, logistic and exponential distributions

Abstract

The term, order statistics, was introduced by Wilks in 1942. However, the subject is much older, as astronomers had long been interested in estimation of location beyond the sample mean. By early 19th century, measures considered included the median, symmetrically trimmed means, the midrange and other related functions of order statistics. In 1818, Laplace obtained (essentially) the distribution of the rth order statistic in random samples and also derived a condition on the parent density under which the median is asymptotically more e cient than the mean. Traditionally, distributions of order statistics have been constructed using the transformation method. Here we used both the transformation method and the new technique of beta generated distributions approach to construct distributions of order statistics. We begin by studying the general properties and functions of order statistics from any continuous distribution. Speci cally, we study the marginal and joint distributions, single and product moments of order statistics as well as distribution of the sample range and median. We then apply these distributional properties of order statistics to the case of uniform, exponential and logistic distributions. Even though, we have used the new technique of beta generated distribution approach in construction of order statistics distributions, we have not discussed this method in detail and we recommend further study on it. Finally, we hope that the knowledge summarized in this study will help in the understanding of order statistics.