Neyman-Pearson Lemma

The Neyman-Pearson lemma is a very useful result in optimization theory. Here we give without proof a statement of this lemma. 

Newton, R., 517, 536,539,560 Newton-Raphson method, 86 Newton s method, 79,80 Newton, T. 0., 555 Neyman-Pearson lemma, 306 Nodal point, 326 Node, 326 proper, 326 Noise... 

There is a well-known theorem in statistics, called the Neyman-Pearson Lemma, which shows that, for a given sample size, it is simply not possible to eliminate these two mistakes we must always trade them off against each another. 

As Talias (2007) has pointed out, there is an interesting analogy between the Pearson Index and the Neyman-Pearson lemma. (The Pearsons in question are different. Alan Pearson is the author of the Pearson index and Egon Pearson, 1895-1980, was the son of Karl Pearson, 1857-1936 and the collaborator of Jerzey Neyman, 1894-1981, in developing hypothesis testing.) Both are relevant to optimizing a function subject to a constraint. In the case of the Pearson index this is profit subject to total cost, and for the Neyman-Pearson lemma it is power subject to the constraint of an overall type I error rate. In both cases a ratio plays a key role. For the Neyman-Pearson lemma this is the likelihood ratio and for the Pearson index the index itself is a ratio of expected profit to expected cost. 

For further discussion of the Pearson index in project management see Burman et al. (2007), Burman and Senn (2003), Regan and Senn (1997), Senn (1996, 1998a) and Senn and Rosati, 2002) for its relation to the Neyman-Pearson lemma see Talias (2007) and for real option theory see Dixit and Pyndick (1994). 

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