Generating the Maximum of Independent Identically Distributed Random Variables

Frequently the need arises for the computer generation of variates that are exact/y distributed as 2 = max(X,, . , X.) where X,, . . . , X, form a sequence of independent identically distributed random variables. For large n the individual generation of the Xi’s is unfeasible, and the inversion-of-a-beta-variate is potentially inaccurate. In this paper, we discuss and compare the corrected inversion method, the log(n)/n-tail method and the record time method. The latter two methods have an average complexity O(logfn)), are very accurate and do not require the inversion of a distribution function. The normal, exponential and gamma densities are treated in detail. The existence of fast and accurate inversion methods for the error function makes the corrected inversion method faster than the other ones for n large enough when the Xi’s are normal random variables.