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Rewriting L(t(~), N)we have |
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Since, asymptotically, q decreases as rapidly as N-1,it is clear that the second term in the brackets will dominate as N grows. Inspecting the earlier expression for L(N) we see the same holds there. Thus, since the second terms are identical |
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 Q.E.D. |
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From this we see that, given the requisite information (µ1, s1) and (µ2, s2), there exist plans which have loss rates closely approximating L*(N)as N increases. |
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The function L*(N)sets a very stringent criterion when there are two uncertain options, specifying a high goal which can only be approached where uncertainty is very limited. Adaptive plans, however, considered in terms of testing schemata, face many more than two uncertain options at any given time. Thus a general performance criterion for adaptive plans must treat loss rates for arbitrary numbers of options. Though the extension from two options to an arbitrary number of r options is conceptually straightforward, the actual derivation of L*(N)is considerably more intricate. The derivation proceeds by indexing the r random variables x1, x2,. . .·,xr so that the means are in decreasing order µ1 > µ2> ... > µr(again, without the observer knowing that this ordering holds). |
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THEOREM 5.3: Under the same conditions as for Theorem 5.1, but now with r random variables, the minimum expected loss after N trials must exceed |
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Proof: Following Theorem 5.1 we are interested in the probability that the average of the observations of any xi, i > 1, exceeds the average for x1. This probabilityof error is accordingly |
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