Estimation of the Scale Parameter of Cauchy Distribution Using Absolved Order Statistics

A new set of ordered random variables generated from a sample from a scale dependent Cauchy distribution known as Absolved Order Statistics (AOS) of the sample forms the problem of investigation in this paper. The distribution theory of these AOS is developed. The vector of AOS is found to be the minimal sufficient statistic for the Cauchy distribution, which is contrary to the existing perception that the vector of order statistics of the sample is minimal sufficient. The best linear unbiased estimate sigma of sigma based on AOS is derived and its variance is also explicitly expressed. Though only n - 4 intermediate order statistics are usable to determine the BLUE of s based on order statistics, it is found that n - 2 AOS are usable to determine <^>s. This makes <^>s a more efficient estimate of s than all of its competitors especially when the sample size is small. Illustration for the above result is made through a real life example. It is found that censoring based on AOS is more realistic and the estimate obtained from it for s is more efficient than the case of censoring with order statistics. A new ranked set sampling called Adjusted Ranked Set Sampling which is suitable for the Cauchy distribution and results in observations distributed as AOS is developed in this paper. Its role in producing a better estimate for s is analyzed.