Read e-book online Large deviations techniques and applications PDF

By Amir Dembo, Ofer Zeitouni

This publication offers an advent to the idea of huge deviations. huge deviation estimates have proved to be the an important instrument required to address many questions in data, engineering, statistial mechanics, and utilized likelihood. the math is rigorous and the functions come from a variety of parts, together with elecrical engineering and DNA sequences. the second one version contains new fabric on focus inequalities and the metric and susceptible convergence ways to giant deviations. basic statements and functions were sharpened, new routines additional, and the bibliography up to date. Amir Dembo is affiliate Professor of arithmetic and records at Stanford collage, and Professor of electric Engineering on the Technion-Israel Institute of know-how. He at the moment serves at the editorial board of the Annals of likelihood. Ofer Zeitouni is Professor of electric Engineering on the Technion-Israel Institute of know-how. He has served at the editorial board of the IEEE Transactions on details conception and presently serves at the editorial board of Stochastic strategies and purposes.

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Example text

Fisher made considerable use of likelihood and conditioning concepts (cf. Fisher (1925, 1934, 1956a)) and came close to espousing the LP in Fisher (1956a), but refrained from complete committment to the principle. Versions of the LP were developed and promoted by Barnard in a series of works (Barnard (1947a, 1947b, 1949)). Likelihood concepts were also employed by a number of other statisticians, cf. Bartlett (1936, 1953). The LP received major notice in 1962, due to Barnard, Jenkins, and Winsten (1962) and Birnbaum (1962a).

2 Axiomatic Development The formal statement of the LP is as follows. FORMAL LIKELIHOOD PRINCIPLE. Consider two experiments E, = {X,, θ, {f 1 }) and E« = (Xos θ» {fg})> where θ is the same quantity in each experiment. Suppose that for the particular realizations xί and x| from E, and E2» respectively, THE LIKELIHOOD PRINCIPLE AND GENERALIZATIONS 27 * xx* ( β ) = « x* ( θ ) 1 2 constant c ( i . e . , f Λ ( x ί ) = c f . (x£) / o r α Π θ ) . T/zerc E v ( E Γ x * ) = Ev(E2,x*). LIKELIHOOD PRINCIPLE COROLLARY.

X^ have been observed, and that v. ,n-l. (u ,n~ ) density. The LP thus says that the evidence about ξ is contained in I (ξ), and if we are stopping the experiment nothing else is needed. However, in deciding whether or not to take another observation, it is obvious that knowledge of v is crucial. If v = 1 it may be desirable to take another observation, but if v = 0 it would be a waste of time (since the measuring instrument is broken). This example is related to a limitation of sufficiency (cf.

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Large deviations techniques and applications by Amir Dembo, Ofer Zeitouni


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