1STAT 702/J702 B.Habing Univ. of S.C. 1STAT 702/J702 October 26th, 2006-Lecture 18-Instructor: Brian HabingDepartment of StatisticsTelephone: 803-777-3578E-mail: [email protected] 702/J702 B.Habing Univ. of S.C. 2Today• Order StatisticsSTAT 702/J702 B.Habing Univ. of S.C. 33.7 – Order StatisticsLet X1, X2, …. Xnbe independent random variables with the same CDF FX(x).The values in order from lowest to smallest are the order statistics X(1), X(2), …. X(n).2STAT 702/J702 B.Habing Univ. of S.C. 4First consider the maximum U=X(n).Note that U≤uif and only if all of the Xi≤u.)()( uUPuFU≤=))()((1uXuXPn≤∩∩≤= L)()(1uXPuXPn≤≤= LSTAT 702/J702 B.Habing Univ. of S.C. 5Taking the derivative we get:)()(1uXPuXPn≤≤= LnXXXuFuFuF )]([)()( == L1)]()[()(−=nXXUuFunfufSTAT 702/J702 B.Habing Univ. of S.C. 6The minimum V=X(1)works similarly:nXVvFvF )](1[1)( −−=1)](1)[()(−−=nXXVvFvnfvf3STAT 702/J702 B.Habing Univ. of S.C. 7This method is a bit messier to use for the other order statistics. Another option is the “differential argument”.The trick to getting the joint p.d.f. directly is try to let our insights into discrete distributions apply to continuous random variables.STAT 702/J702 B.Habing Univ. of S.C. 8And we get…knkXkkkXkXxFxFxfknknxfk−−−⋅−−=)](1)[()()!()!1(!)()()(1)()()(STAT 702/J702 B.Habing Univ. of S.C. 9Example 1) Say you conduct 10 independent tests of hypotheses. How small should the smallest p-value be for you to reject it at a 0.05 level? That is, what is the 5th-%ile for the 1storder statistic?4STAT 702/J702 B.Habing Univ. of S.C. 10Example 2) x(k)for a uniform random variable.This is a beta distribution with parameters k and n-k+1.STAT 702/J702 B.Habing Univ. of S.C. 11STAT 702/J702 B.Habing Univ. of S.C. 125STAT 702/J702 B.Habing Univ. of S.C. 13The joint p.d.f. of all of the order statistics is:)()(!),()()1()()1(,)()1(nnXXxfxfnxxfnLKK=STAT 702/J702 B.Habing Univ. of S.C. 14One way to find the joint p.d.f. of a pair of order statistics would be to integrate out the n -2 you are not concerned with.Another way is to use what the “differential argument”.Say we want the joint p.d.f. of X(i)and X(j)where i<j.STAT 702/J702 B.Habing Univ. of S.C. 15And