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September, 1953 Sequential Decision Problems for Processes with Continuous Time Parameter Problems of Estimation
A. Dvoretzky, J. Kiefer, J. Wolfowitz
Ann. Math. Statist. 24(3): 403-415 (September, 1953). DOI: 10.1214/aoms/1177728980


In a recent paper [1] the authors began the study of the theory of sequential decision functions for stochastic processes with a continuous time parameter. This paper treated the standard problem of testing hypotheses, and the advantage of being able to stop at an arbitrary time point (not necessarily a multiple of some unit given in advance) was demonstrated in several cases, notably in that of deciding between two Poisson processes. The optimal tests were Wald probability ratio tests and thus truly sequential. In the present paper we treat the problem of estimation, and study in detail the Poisson, Gamma, Normal and Negative Binomial processes. It turns out for these processes that, with a proper weight function, the minimax (sequential) rule reduces to a fixed-time rule. Though we confine ourselves to point-estimation it is clear that similar methods apply to interval estimation. It may also be remarked that the case when the time-parameter is discrete need not be treated separately. For example, as described in Section 6.1, the results of Sections 2 and 3 imply analogous results in the case of discrete time, which in turn imply certain results proved in [3] and (in the nonsequential case) in [2] by other methods. The treatment of some other problems in estimation is discussed in Section 6. This paper may be read independently of [1].


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A. Dvoretzky. J. Kiefer. J. Wolfowitz. "Sequential Decision Problems for Processes with Continuous Time Parameter Problems of Estimation." Ann. Math. Statist. 24 (3) 403 - 415, September, 1953.


Published: September, 1953
First available in Project Euclid: 28 April 2007

zbMATH: 0051.36606
MathSciNet: MR57523
Digital Object Identifier: 10.1214/aoms/1177728980

Rights: Copyright © 1953 Institute of Mathematical Statistics


Vol.24 • No. 3 • September, 1953
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