Statistics Research Letters (SRL) Volume 3, 2014 www.srl-journal.org L-Moments, TL-Moments Estimation and Recurrence Relations for Moments of Order Statistics from Exponentiated Inverted Weibull Distribution Jagdish Saran *1 , Devendra Kumar 2 , N. Pushkarna *1 , Rashmi Tiwari *1 *1 Department of Statistics, University of Delhi, Delhi-110007, India 2 Department of Statistics, Amity Institute of Applied Sciences, Amity University, Noida-201 303, India *1 jagdish_saran52@yahoo.co.in; 2 devendrastats@gmail.com Abstract In this paper we have established exact expressions and some recurrence relations for single and product moments of order statistics from exponentiated inverted Weibull distribution. Further the characterization of this distribution has been considered on using a recurrence relation for single moments. The first four moments and variances of order statistics are computed for various values of parameters α and β . We have also obtained L-moments and TL-moments of the above distribution and used them to find the L- moments and TL-moments estimators of the parameters α and β of the distribution. Keywords Order Statistics; Record Values; Exponentiated Inverted Weibull Distribution; Single and Product Moments; Recurrence Relations; L-Moments; TL-Moments and Characterization Introduction If random variables 1 2 , , , n X X X  are arranged in ascending order of magnitude such that 1: 2: : : n n rn nn X X X X ≤ ≤ ≤ ≤ ≤   , then : rn X is called the r − th order statistic. 1: 1 2 min ( , , , ) n n X X X X =  and : 1 2 max ( , , , ) nn n X X X X =  are called extreme order statistics or the smallest and the largest order statistics. The subject of order statistics deals with the properties and applications of these ordered random variables and of functions involving them (David and Nagaraja, 2003). Asymptotic theory of extremes and related developments of order statistics are well described in an applausive work of Galambos (1987). Also, references may be made to Sarhan and Greenberg (1962), Balakrishnan and Cohen (1991), Arnold et al. (1992) and the references therein. It is different from the rank order statistics in which the order of the value of observation rather than its magnitude is considered. It plays an important role both in model building and in statistical inference. For example: extreme values are important in oceanography (waves and tides), material strength (strength of a chain depends on the weakest link) and meteorology (extremes of temperature, pressure, etc). Let 1 2 , , , n X X X  be a random sample of size n from a continuous probability density function ( ) pdf () fx and the distribution function ( ) df () Fx . Then the pdf of : rn X , 1 r n ≤ ≤ , is given by 1 : : () [ ( )] [1 ( )] () r n r rn rn f x C Fx Fx fx − − = − , x −∞ < <∞ , (1.1) and the joint pdf of : rn X and : sn X , 1 r s n ≤ < ≤ , is given by 1 1 ,: ,: (, ) [ ( )] [ () ( )] r s r rsn rsn f xy C Fx Fy Fx − −− = − [1 ( )] () () n s Fy fxf y − × − , x y −∞ < < <∞ , (1.2) where 1 : ! [ (, 1)] ( 1)!( )! rn n C Brn r r n r − = = − + − − and 1 ,: ! [ (, , 1)] ( 1)!( 1)!( )! rsn n C Brs rn s r s r n s − = = − − + − − − − . Several recurrence relations between raw and central moments of order statistics from different distributions are available in the literature. The main utility and advantage of such recurrence relations 63