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

Order Statistics: Let X1, . . . , Xn be i.i.d. from a continuous distribution F, and let X(i) denote the ith smallest of X1, . . . , Xn, 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.

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(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 βi 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

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