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where ni is the number of trials given xi, and the loss ranges from (µ1 - µ2) to (µ1 - µr) depending on which xi is mistakenly taken for best. |
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Let  , let m = min {n2, n3, . . . , nr}, and let j be the largest index of the random variables (if more than one) receiving m trials. |
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The proof of Theorem 5.1 shows that a lower bound on the expected loss is attained by minimizing with respect to any lower bound on the probability q (a point which will be verified in detail for r variables). In the present case q must exceed |
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using the fact that (µ1 - µr) ³ (µ1 - µj) for any j > 1. By the definition of q |
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using the fact that (µ1 - µ2) £ (µ1 - µi) for i ³ 2. Moreover the same value of n minimizes both LN,t(n) and L'N,t(n). To find this value of n, set |
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Solving this for n* and noting that 1 - 2q rapidly approaches 1 as N increases, gives |
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Noting that q must decrease less rapidly than q' with increasing n, we have (dq'/dn) < (dq/dn) and, taking into account the negative sign of the derivatives, |
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(This verifies the observation at the outset, since the expected loss approaches n* as N increasessee below.) Finally, noting that n > (r - 1)m, we can proceed as in the two-variable case by using (r - 1)m in place of n and taking the derivative of q' with respect to m instead of n. The result is |
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