Marwa A. A et al Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 4( Version 1), April 2014, pp.77-90 www.ijera.com 77 | Page Comparison of Estimators for Exponentiated Inverted Weibull Distribution Based on Grouped Data Amal S. Hassan 1 , Marwa A. A 1 , Hegazy Zaher 1 and E. A. Elsherpieny 1 ( 1 Department of Mathematical Statistics, Institute of Statistical Studies& Research, Cairo University, Orman, Giza, Egypt) ABSTRACT In many situations, instead of complete sample, data is available only in grouped form. This paper presents estimation of population parameters for the exponentiated inverted Weibull distribution based on grouped data with equi and unequi-spaced grouping. Several alternative estimation schemes, such as, the method of maximum likelihood, least lines, least squares, minimum chi-square, and modified minimum chi-square are considered. Since the different methods of estimation didn't provide closed form solution, thus numerical procedure is applied. The root mean squared error resulting estimators used as comparison criterion to measure both the accuracy and the precision for each parameter. Keywords - exponentiated inverted Weibull, grouped data, least lines, least squares, maximum likelihood, minimum chi square, modified minimum chi square. I. INTRODUCTION In different fields of science such as engineering, biology, and medicine it is not possible to obtain the measurements of a statistical experiment correctly but is possible to classify them into intervals, rectangles or disjoint subsets [Alodat and Al-Saleh [1]; Shadrouk and Nasiri [2]]. This means that a raw sample grouped into a frequency distribution with equi or unequi-spaced intervals, for example in life testing experiments, the failure time of a component is observed to the nearest hour, day or month. In this case the values of individual observations are not known, but the number of observations that fall in each group is only known. Exact measurements often require costly skilled, personnel and complex instruments; whereas grouped data is usually quicker, easier and cheaper. The estimation problem of the unknown parameters from different distributions based on grouped samples has considered by many authors. Earlier works on grouped samples can be found in Kulldorff [3], who got the maximum likelihood estimate (MLE) of the exponential distribution parameter for grouped data. Some other works can be found in, see for example, Archer [4], Cheng and Chen [5], Rosaiah et al. [6], Rao et al [7], Chen and Mi [8], Komori and Hirose [9] , Kantam et al. [10]. More recently, Shadrokh and Pazira [11] obtained the classical and Bayesian estimation from grouped and un-grouped data when the underlying distribution is exponentiated gamma. Shadrokh and Nasiri [12] considered the ungrouped and grouped data problems for the minimax distribution. Marwa et al. [13] obtained the MLE of the unknown parameter for the exponentiated Fréchet distribution based on grouped data. Also, asymptotic optimum group limits in the case of unequi-spaced groupings is worked out. Adding one or more parameters to a distribution makes it richer and more flexible for modeling data. There are different ways for adding parameters to a distribution. Exponentiated (generalized) inverted Weibull distribution is a generalization to inverted Weibull through adding a new shape parameter by exponentiation to distribution function. The standard two-parameter exponentiated inverted Weibull distribution (EIW) distribution has been proposed by Flaih et al. [14] with the following cumulative distribution function (CDF)     ) ( ) , ; (    x e x F ; x ,  ,  >0 (1) which is simply the  -th power of the distribution function of the standard inverted Weibull distribution with two shape parameters  and .  Therefore, the probability density function(PDF) is:        ) ( ) , ; ( ) 1 (      x e x x f ; x >0 (2) For  =1, it represents the standard inverted Weibull distribution, and for  = 1 it represents the exponentiated standard inverted exponential distribution. The exponentiated inverted Weibull distribution has been studied by a few authors, for example; Hassan [15] concerned with the optimal designing RESEARCH ARTICLE OPEN ACCESS