Suppose n balls having weights w1, w2, . . . , wn are in an urn. These balls are

Question

Suppose n balls having weights w₁, w₂, . . . , w_n are in an urn. These balls are sequentially removed in the following manner: At each selection, a given ball in the urn is chosen with a probability equal to its weight divided by the sum of the weights of the other balls that are still in the urn. Let I₁, I₂, . . . , I_n denote the order in which the balls are removed—thus I₁, . . . , I_n is a random permutation with weights.

(a) Give a method for simulating I₁, . . . , I_n.
(b) Let X_i be independent exponentials with rates w_i , i = 1, . . . , n. Explain how X_i can be utilized to simulate I₁, . . . , I_n.

Accepted Answer

(a) They can be simulated in the same sequential fashion in which they are defined. That is, first generate the value of a random variable I₁ such that

[latex]P\left\{I_{1}=i ight\}=\frac{w_{i}}{\sum_{j=1}^{n} w_{j}}, \quad i=1, \ldots, n[/latex]

Then, if I₁ = k, generate the value of I₂ where

[latex]P\left\{I_{2}=i ight\}=\frac{w_{i}}{\sum_{j eq k} w_{j}}, \quad i eq k[/latex]

and so on. However, the approach given in part (b) is more efficient.
(b) Let I_j denote the index of the j th smallest X_i

Question 11.EX.16: Suppose n balls having weights w1, w2, . . . , wn are in an ......

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