Numerical Computation of Stochastic Inequality Probabilities

This paper addresses the problem of numerically evaluating P(X > Y ) (1) for independent continuous random variables X and Y . This calculation arises in the design of clinical trials and as such appears in the inner loop of simulations of these trials. An early example of this is given by Thompson (1933), with more recent examples by Giles et al (2003), and Berry (2003, 2004). It is worthwhile to optimize the calculation of these probabilities as they may be computed millions of times in the course of simulating a single trial. Techniques such as memoization (Orwant 2002) can eliminate redundant calculations of such probabilites throughout a simulation, but the need for a large number of evaluations remains. After considering how to compute (1) in general, we present optimizations for important special cases in which X and Y both belong to one of the following families of classical distributions: exponential, gamma, inverse gamma, normal, Cauchy, beta, and Weibull. Numerical Computation of Stochastic Inequality Probabilities John D. Cook Division of Quantitative Sciences, Unit 1409 The University of Texas, M. D. Anderson Cancer Center P. O. Box 301402, Houston, Texas 77030, USA [email protected] First published December 10, 2003. This revision August 8, 2008. Abstract This paper addresses the problem of numerically evaluating P (X > Y ) (1) for independent continuous random variables X and Y . This calculation arises in the design of clinical trials and as such appears in the inner loop of simulations of these trials. An early example of this is given by Thompson (1933), with more recent examples by Giles et al (2003), and Berry (2003, 2004). It is worthwhile to optimize the calculation of these probabilities as they may be computed millions of times in the course of simulating a single trial. Techniques such as memoization (Orwant 2002) can eliminate redundant calculations of such probabilites throughout a simulation, but the need for a large number of evaluations remains. After considering how to compute (1) in general, we present optimizations for important special cases in which X and Y both belong to one of the following families of classical distributions: exponential, gamma, inverse gamma, normal, Cauchy, beta, and Weibull.