A variation of the importance sampling algorithm of Exercise 5.64 can actually produce an approximate sample from f. Again let X ~ f and generate Y1, Y2,..., Ym, iid from g. Calculate qi = [(Yi)/g(Y)]/[∑mj=1 f(Yj)/g(Yj)}. Then generate random variables X* from the discrete distribution on Y1, Y2,..., Ym, where P(X" = Yk) = qk. Show that X*1, X*2,..., XT* is approximately a random sample from f.
(Show that P(X* ≤ x) = ∑mj=1 qiI(Yi ≤ x), let m →∞ , and use the WLLN in the numerator and denominator.)
This algorithm is called the Sampling/Importance Resampling (SIR) algorithm by Rubin (1988) and is referred to as the weighted bootstrap by Smith and Gelfand (1992).