On the Theory of Apportionment
On the Theory of Apportionment
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论分配理论
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发表时间：
1935
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通讯作者：
W. R. Thompson
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作者：
W. R. Thompson
1. 如果在可接受的意义上，P是一种处理方法T优于另一种处理方法T2的概率，那么我们可以开发一种分配系统，使T1的比例使用是f(P)，这是一个单量递增函数，而不是在某一点上完全不进行区分，然后最终完全拒绝其中一个。据我所知，迄今为止在他的领域发表的唯一一篇论文是我在最近一期的《bioometrilca》上发表的一篇论文。在本文中，我考虑了在两个这样的竞争处理之间进行选择的情况，对于对称性，建议f(QI f(p)，其中Q = 1 p。然后，当它不是更好时，分配给T1的风险是Q f(p)，而T2的相应风险是p f(Q)。因此，我进一步建议我们设置f(p) p，这是这两个风险相等的充分必要条件。它们的和，总风险q，是2PQ。考虑了一种特殊情况，其中在任何给定的试验中使用Ti的结果是成功或失败，失败的概率是未知的，pi，先验的（独立于i 1，， kc），同样可能位于可能范围（0,1）中的任何两个相等间隔中的任何一个。进一步假设，对于给定的Ti，我们有恰好ni个独立试验的经验，su？访问次数为si，失败次数为ri = si-；得到这样一个样本的概率是
1. If in an accepted sense, P is the probability that one method of treatment, T,, is better than a rival, T2, we may develop a system of apportionment such that the proportionate use of T1 is f(p), a monotolle increasing functioni, rather than make no discriminationi at all up to a certaini poinit and then finally entirely reject one or the other. The only paper * which has so far appeared in his field, as far as I am aware, is one by myself in a recent issue of Biometrilca. In this paper I have considered the case of choice between two such rival treatments,t and for symmetry suggested that f (QI f(p) where Q = 1 P. Then the riski of assignmenit to T1 when it is not the better is Q f(p), while the correspondinig risk for T2 is P f.(Q Accordingly, I suggested further that we set f(p) P, which is a necessary and sufficient conditioni that these two risks be equal. Their sum, the total qrisk, is then 2PQ. A special case was considered wherein the result of use of Ti at any given trial is either success or failure, the probability of failure being an unkniownl, pi, a priori (independenitly for i 1, , kc) equally likely to lie in either of any two equal intervals in the possible range, (0, 1). It is further assumed that for a given Ti we have an experience of exactly ni indepenidenit trials, the number of su?ccesses being si and of failu-res being ri = si-; and the probability of obtaininig such a sample is