Generating Samples

Abstract

Generating a random sample constitutes an essential feature of every Monte Carlo experiment. If

$$\zeta = \int_R {\varphi (x)} dx,$$

(1)

as in expression (2.67), is to be estimated, then a procedure that randomly generates points from the uniform distribution on ℱ^m would suffice. When the cost of evaluating φ(x) in Algorithm ${\overline \zeta _n}$ in Sec. 2.7 is prohibitively high, the equivalence of expression(1) and

$$\zeta = \int_{{\mathbb{R}^m}} {\kappa (Z)dF(Z)}$$

(2)

in expression (2.72c) becomes of interest. Recall that, for A⊑ℬ ⊑ℝ^m, 0≤F(A) ≤F≤ F(ℬ) ≤F(ℝ^m) = 1. Since F is a distribution function, an alternative procedure randomly samples Z ⁽¹⁾,...,Z ⁽ⁿ⁾ independently from F and computes

$${\ddot \zeta _n} = {n^{ - 1}}\sum\limits_{i = 1}^n {\kappa ({Z^{(i)}})}$$

as an unbiased estimate of ζ. Although generating Z ⁽ⁱ⁾ from F always costs more than generating X ⁽ⁱ⁾ from the uniform distribution on ℱ^m, evaluating k(Z ⁽ⁱ⁾) may be considerably less costly than evaluating φ(X ⁽ⁱ⁾).