# Order Statistics: Let X1, . . . , Xn be i.i.d. from a continuous distribution F

Order Statistics: Let X_{1}, . . . , X_{n} be i.i.d. from a continuous distribution F, and let X(i) denote the ith smallest of X_{1}, . . . , X_{n}, i = 1, . . . , n. Suppose we want to simulate X_{(1)} < X_{(2)} < ··· < X_{(n)}. One approach is to simulate n values from F, and then order these values. However, this ordering, or sorting, can be time consuming when n is large.

(a) Suppose that λ_{(t)}, the hazard rate function of F, is bounded. Show how the hazard rate method can be applied to generate the n variables in such a manner that no sorting is necessary.

Suppose now that F^{−1} is easily computed.

(b) Argue that X_{(1)}, . . . , X_{(n)} can be generated by simulating U_{(1)} < U_{(2)} < ···<U_{(n)}—the ordered values of n independent random numbers—and then setting X_{(i)} = F^{−1}(U_{(i)}). Explain why this means that X_{(i)} can be generated from F^{−1}_{(βi)} where β_{i} is beta with parameters i, n +i +1.

(c) Argue that U_{(1)}, . . . , U_{(n)} can be generated, without any need for sorting, by simulating i.i.d. exponentials Y1, . . . , Yn+1 and then setting