How to Simulate Multivariate Distributions?

Abstract

Before we start, let us note that we use ‘to simulate’ and ‘to sample’ as synonyms throughout. The present chapter (i) provides a motivation for the simulation of random vectors, and (ii) shows how this can be achieved. A probability law is a priori a purely analytical object, that is, a function which gets sets as input and returns numbers in [0, 1]. We interpret the input sets as events that might happen or not, and the output as probabilities corresponding to the likelihood of occurrence of the input events. For example, the uniform distribution u[a, b] assigns the value (y − x)/(b − a) to the interval [x, y] for a ≤ x y ≤ b. In most applications, we work with a parametric family of probability laws, that is, the likelihood of some event can be altered by changing the parameters of the probability law. For instance, the two-parametric family u[a, b] of uniform distributions assigns the value 0.5to the set [0,1] if a = 0 and b = 2; and it assigns the value 0.25 to the same set [0,1] if the parameters are changed to a=−1 and b = 3.