Applied Statistical Decision Theory by Howard Raiffa

By Howard Raiffa

"In the sector of statistical determination concept, Raiffa and Schlaifer have sought to advance new analytic innovations through which the trendy thought of application and subjective likelihood can truly be utilized to the industrial research of ordinary sampling problems."
—From the foreword to their vintage paintings Applied Statistical selection Theory. First released within the Sixties via Harvard collage and MIT Press, the e-book is now provided in a brand new paperback variation from Wiley

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Our aim is to describe an interaction of structural properties of the space X and the function P0 (·) which are enough for P0 (·) to retain this determining character without the strong probabilistic assumptions of the Poisson process. Concerning terminology, Kendall (1974) used the term avoidance function in a more general (stochastic geometry) context, reflecting the fact that P0 (A) gives the probability of the support of a random set function avoiding a prescribed set A; other possible terms include zero function, avoidance probability function, and vacuity function [McMillan (1953)].

Denoting distinct points 34 9. Basic Theory of Random Measures and Point Processes of X , we should have for the first few fidi distributions P1 ({i}; 0) = 1 − P1 ({i}; 1) = P0 ({i}), P2 ({i}, {j}; 0, 0) = P0 ({i, j}), P2 ({i}, {j}; 0, 1) = P0 ({j}) − P0 ({i, j}), P2 ({i}, {j}; 1, 1) = 1 − P0 ({i}) − P0 ({j}) + P0 ({i, j}). Continuing in this way, all the fidi distributions could be built up through a sequence of differencing operations applied to P0 (·), and it is clear that the avoidance function would thereby determine the fidi distributions uniquely.

Xk ). Proof. VI(a) and therefore necessary. We show that it is also sufficient. Let us first point out how the extension from disjoint to arbitrary families of sets can be made. Let {B1 , . . , Bn } be any such arbitrary family. Then there exists a minimal family {A1 , . . , Ak } of disjoint sets (formed from the nonempty intersections of the Bi and Bic ) such that each Bi can be represented as a finite union of some of the Aj . The joint distribution Fk (A1 , . . , Ak ; x1 , . . , xk ) will be among those originally specified.

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