17. Order Statistics: Let be i.i.d. from a continuous distribution F, and let denote the ith smallest of

. Suppose we want to simulate

. 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 , 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 is easily computed.

(b) Argue that can be generated by simulating Image—the ordered values of n independent random numbers—and then setting Image. Explain why this means that can be generated from Image where is beta with parameters Image.

(c) Argue that can be generated, without any need for sorting, by simulating i.i.d. exponentials Image and then setting