International Journal of Statistika and Mathematika ISSN: 2277- 2790 E-ISSN: 2249-8605, Volume 8, Issue 3, 2014 pp 102-104 International Journal of Statistiika and Mathematika, ISSN: 2277- 2790 E-ISSN: 2249-8605, Volume 8 Issue 3 Page 102 Selection of the Best New Better than used Population Based on Subsamples Sujatha Inginshetty 1* , Parameshwar V. Pandit 2# 1 Department of Statistics, Gulbarga University, Gulbarga – 585106, Karnatka, INDIA. 2 Department of Statistics, Bangalore University, Banglore – 560056, Karnatka, INDIA. Corresponding Addresses: * sujatasi_gug@rediffmail.com , # panditpv12@gmail.com Research Article Abstract: The present study considers the problem of selecting the ‘Best’ new better than used(NBU) population among the several NBU populations. The procedure to select the ‘Best’ NBU population is developed based on a measure of departure from exponentiality towards NBU, proposed by Pandit and Math(2009) for the problem of testing exponentiality against NBU alternatives in one sample setting. The selection procedure is based on large sample properties of the statistic proposed in Pandit and Math(2009).We also indicate some applications of the selection procedure. Keywords and Phrases: New better than used(NBU), Selection and ranking, U-statistic, Probability of correct selection. 1. Introduction The selection of the ‘best population’ among the several populations is based on the principle of ranking and selection developed in the literature. One of the criteria of selection is to select a population that belongs to some parametric family with reference to a parameter. Another approach is to select a population that belongs to Lehmann’s (1963) procedure based on ranks, Barlow and Proschan’s (1969) approach based on partial orderings of probability distributions. Further, a selection procedure based on means for selecting from class of increasing failure rate(IFR) distributions is given by Patel (1976). An extensive review of ranking and selection problems can be found in Gupta and Panchapakesan(1979). Here, we consider selection problem of selection for the NBU classes of distributions. Pandit and Math (2009) gave  ∞       = 0 ) ( ) ( x dF m x F F γ as a measure of NBU -ness and used it to develop a test for testing exponentiality against NBU distributions. When F is exponential, 1 1 ) ( + = m F γ and if F is NBU 1 1 ) ( + ≥ m F γ . It is easy see that, if ) ( ) ( G F γ γ ≥ , then a life distribution F is said to possess more NBU -ness property than that of another life distribution G, Here, the criterion for selecting the best distribution possessing NBU –ness property among k NBU populations is based on the value of ) ( F γ , assuming that the underlying distribution F is continuous. The selection procedure here is based on the U- statistics T n, (F), an unbiased estimate of ) ( F γ which is defined as below: Let X 1 ,X 2 , … ,X n be a random sample from F. Define  <         = i j j i n X X h n F T ) , ( 3 1 ) ( where ) , ( j i X X h is the symmetrized version of         > k X X I j i and    ∈ = otherwise A x if A I 0 1 ) ( . Here summation is taken over all         2 n combinations of integers ) , ( j i taken out of integers{1,2,3,…,n}. In this paper, the asymptotic normality of T n (F) ( Refer Pandit and Math (2009) ) is used for selecting the most NBU distribution among k NBU distributions. 2. Selection procedure Let F 1, F 2, …, F k be the distribution functions of k NBU distributions and hence the functional form of F i and the NBU -ness measure ) ( i F γ of F i are unknown, but it is assumed that F i are continuous. For the sake of convenience, let ) ( i F γ be denoted by i γ . The goal is to select the population which has largest [] k γ , where [] [] [] k γ γ γ ≤ ≤ ≤ . .......... 2 1 denote the ordered NBU -ness measure for k distributions. Selecting largest [] k γ refers to selecting the most NBU distribution. Let [] [] [] ( ) k γ γ γ γ ..... , , 2 1 = - and [] [] []       ≤ ≤ ≤ ≤ ≤ + = Ω - - 1 ...... . . . 1 1 : 2 1 k m γ γ γ γ be the parameter space which is partitioned into a preference zone ( ) * δ Ω brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by ePrints@Bangalore University