Rank order probabilities: one-sample normal shift alternatives. Part A: table of rank order probabilities
OAK RIDGE NATIONAL LAB., TENN. MICHIGAN STATE UNIV., EAST LANSING. CYCLOTRON LAB., C.S. Lever · 1973
Let Xi, ..., Xv be independent random variables normally distributed with mean 9 and variance 1.The normal density function will be denoted by f(x, 0).If Xi. .... XJJ are the observations and yi...., y^ are the absolute values of the observations arranged from smallest to largest, then z = (z^, .... z^) is defined to be the observed rank order where *j = 1 if y* is the absolute value of a positive number and z^ = 0 if y* is tne absolute value of a negative number.Corresponding to z is the random vector Z.There are 2^ possible values of Z and f( Ui -(2z t -l)e)du i .Tables of P(Z = z|0) to nine decimal places are presented for all z for 1 has wei ^ht V( 2 -5! l!).Examination of the approximation to the integral suggests that evaluation requires on the order of M arithmetic operations.However.Milton and I. R. Savage developed an algorithm which provides an efficient method of calculating the approxi mation by making use of the factorization of the integrand.The approxi mation converges to the true value of the integral as the number of subintervals M is increased.This midpoint approximation is a product of midpoint approximations for single dimensional integrals.Many quadrature formulas for numerical integration of multiple integrals are not valid for regions other than for cubes or parallelopipeds.Haber .p. ^86.However, quadrature formulas that are products of the formulas for each dimension may be used for other regions.Haber, p. 189-rt The algorithm is described in terms of the vectors £.. v .. and £.. j = l, ...,]f,l 1: for j = 2 N and k = 1, .... J-l.calculate the following vectors:With this definition, we obtain Theorem 3.1.N Theorem ?.l: The midpoint approximation to I" is I = N! h s" __. .where « N » lykl i« the (M*l)st element of s^..As h -0, N! h a-_ --I".Note: In applying the theorem vhen the original region of integration is 0 < x. < x« < ... <!_<•, one must simultaneously allow b to diverge as h -o.Proof: It is possible to show that the approximation given by the algorithm is the same as the approximation given by (5-2).That is, it can be shown that N l 8.50 and 8 up to 1.5.f(x-0) < 1.0 x lo" n .(For 0 = 2.0 and 0 = 3.0, the range of calculations was extended by setting b = 10.00.)