Monte Carlo simulation and its statistical design and analysis

Abstract

In this paper we shall discuss statistical aspects of the numerical technical called Monte Carlo simulation. In the present section we examine Monte Carlo simulation in general. Any solution method using random numbers (or pseudorandom numbers) we call a Monte Carlo method. As we know random numbers are stochastic variables that are uniformly distributed on the interval [0.1] and are statistically independent. On digital computers random numbers are approximated by pseudorandom numbers, i.e. numbers generated by deterministic algebraic formulas and for practical purposes considered to be purely random. Presently, the most used formula is the multiplicative congruential one, i.e. x i = a x i-1 + b (mod m) for i = 1, 2, ..., with prespecified a,b, m and x 0 . It is recommended to permute the resulting (pseudo) random numbers, r i = x i /m, in order to improve their statistical independence. Numerous publications on random number generation are available. For briefness' sake we mention only the bibliography (with 491 references) by Nance and Overstreet (1972): Marsaglia (1972) giving a table of recommended values for a when m = 2 32 , 2 35 , 2 36 ; and Lewis (1972). Transformations of random numbers can yield any other stochastic variable. For recent results, see Ahrens and Dieter (1972), Andrews et al. (1972, p. 56), Mihram (1972, p. 94--146), Nance and Overstreet (1972), Newman and Odell (1971, p. 18--52).