ELSEVIER Computational Statistics & Data Analysis 29 (1998) 35-53 COMPUTATIONAL STATISTICS & DATA ANALYSIS Best- and worst-case variances when bounds are available for the distribution function George S. Fishman *, David S. Rubin Department of Operations Research, University of North Carolina, Chapel Hill. NC 27599, USA Received June 1997; received in revised form March 1998 Abstract Consider a discrete random variable, X, taking on values {I,2 ..... t} and with lower and upper bounds on its unknown distribution function (d.f.). Within these constraints, this paper derives the forms of the d.f.'s that lead to the largest possible (worst-case) variance and to the smallest possible (best-case) variance for any monotone function of X. The paper also describes Algorithms NLPW and NLPB for computing these worst- and best-case variances, each in O(t) time. A network reliability example illustrates how these techniques can be used to bound sample size in a Monte Carlo experiment. (~) 1998 Elsevier Science B.V. All rights reserved. For lower and upper bounds on the distribution function (d.f.) of a discrete random variable, this paper shows how to compute the induced best-case and worst-case bounds on the variance for a monotone function of the random variable. Let X be a discrete random variable taking on the values i = 1,2,...,t and let n:= {it,>0; i= 1,2,...,t} be its unknown probability mass function (p.m.f.). Let w J:= {Wlj, W2j,...,Wtj } for j= 1,2 .... ,J be monotone decreasing sequences for which the J unknown expectations t ~j(~z)= Zwijni, j= 1,2 ..... J i=1 " Corresponding author. 0167-9473/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PII S0167-9473(98)00051-6