Optimal Sequentially Planned Decision Procedures by Norbert Schmitz (auth.), Norbert Schmitz (eds.)

By Norbert Schmitz (auth.), Norbert Schmitz (eds.)

Learning from event, making judgements at the foundation of the to be had info, and continuing step-by-step to a wanted objective are basic behavioural traits of people. however, it was once no longer until eventually the early 1940's that the sort of statistical concept - particularly Sequential research - was once created, which permits us to enquire this type of behaviour in an exact demeanour. A. Wald's well-known sequential chance ratio attempt (SPRT; see instance (1.8» grew to become out to have a big impact at the improvement of this conception. at the one hand, Wald's primary monograph "Sequential research" ([Wa]*) is largely based round this attempt. however, very important houses of the SPRT - e.g. Bayes­ optimality, minimax-properties, "uniform" optimality with recognize to anticipated pattern sizes - gave upward push to the advance of a normal statistical determination concept. As a conse­ quence, the SPRT's performed a dominating function within the additional improvement of sequential research and, extra commonly, in theoretical statistics.

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EZTi in particular EZTtJ = v. t. Mil =F 0 and all j E M" the sets B(aj) belong to BIl . t. Mb =F 0,j E M b. e. r: E T" Va E A. L bEA:bta VaEA EZb < 00, (ii)j(iii) are shown by backward induction: For each a E E(ZT! t. s. e. t. tg(b). A - in particular for Consider now i E {l, ... t. : a be a sampling plan. : ak. For C E B" follows [e ZT J( dP = [ Jen{T="} ZT dP +L [ hEM. Jen{Tt"h} ZT dP = Jen{T="} [ Z" dP + L [ Z"'k dP hEM. } Z" dP + kEM. L Jen{Ttlllc} [ E(E(Z.... I Bilk) I B .. : ak} E BIl and B ..

4 Optimal sampling plans for the Markov case The previous results may be considerably simplified if the pay-off process (Z.. ).. eA has a "Markov structure" (defined in the sequel) which has as a consequence that the future development of the process at each decision time depends only on the present "state" but not on the "way" this state was reached. We will show that one may, in these cases, restrict one's attention to sampling plans which take into consideration only the current "state" of the process.

22) need no further explanation. 23) Remarks a) Quite similar to the purely sequential case the u-algebra B.. represents the information available due to the sample sequence a. b) The B.. -measurability assumptions on sequential sampling plans again have the meaning that the decision whether to stop with the sample sequence or what size the next (sub-) sample should have may depend only on the information gathered up to that stage. -measurability of the terminal decision functions

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