On Chernoff's Hypotheses Testing Problem for the Drift of a Brownian Motion

Zhitlukhin, Mikhail; Muravlev, Alexey Anatol'evich

doi:10.1137/s0040585x97986266

Cited by 9 publications

(14 citation statements)

References 7 publications

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“…where T T −t denotes the set of stopping times less or equal to T − t with respect to the completed filtration of {Π t,π t+s } s≥0 . Note that all results for the perpetual problem (22) described above in this section also hold for the finite horizon problem (35), with the obvious modifications regarding the time horizon, by the same proofs. Moreover, the pay-off process in (35) is continuous and bounded, so standard optimal stopping theory (see, for example, [20, Corollary 2.9 on p. 46]) yields that τ T := inf{s ≥ 0 : Π t,π t+s / ∈ (b T 1 (t + s), b T 2 (t + s))} is an optimal stopping time in (35), where b T 1 and b T 2 are the corresponding boundaries enclosing the finite-horizon continuation region…”

Section: 2mentioning

confidence: 59%

“…Since v is continuous, C is open and D is closed. Resorting to intuition from optimal stopping theory, we expect that the stopping time τ * := inf{s ≥ 0 : (t + s, Π t,π t+s ) ∈ D} (26) is an optimal stopping time in (22). (Note that standard optimal stopping theory does not apply since the pay-off process is not uniformly integrable.)…”

Section: 2mentioning

confidence: 99%

“…The stopping times τ T τ * a.s. as T ∞, where τ * is defined in (26). Moreover, τ * is optimal in (22).…”

Section: 2mentioning

confidence: 99%

“…However, then the corresponding boundaries are not necessarily monotone, so the corresponding free-boundary problem is less tractable. For Chernoff's loss function and the particular case of a normal prior distribution, [22] finds an appropriate scaling and time-change of X in that the transformation guarantees the monotonicity of the corresponding stopping boundaries. The extension to more general prior distributions and other loss functions remains an important and challenging open problem.…”

Section: Introductionmentioning

confidence: 99%

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Bayesian sequential testing of the drift of a Brownian motion

Ekström

Vaicenavicius

2015

ESAIM: PS

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We study a classical Bayesian statistics problem of sequentially testing the sign of the drift of an arithmetic Brownian motion with the 0-1 loss function and a constant cost of observation per unit of time for general prior distributions. The statistical problem is reformulated as an optimal stopping problem with the current conditional probability that the drift is non-negative as the underlying process. The volatility of this conditional probability process is shown to be non-increasing in time, which enables us to prove monotonicity and continuity of the optimal stopping boundaries as well as to characterize them completely in the finite-horizon case as the unique continuous solution to a pair of integral equations. In the infinite-horizon case, the boundaries are shown to solve another pair of integral equations and a convergent approximation scheme for the boundaries is provided. Also, we describe the dependence between the prior distribution and the long-term asymptotic behaviour of the boundaries.

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Section: 2mentioning

confidence: 59%

Section: 2mentioning

confidence: 99%

“…The stopping times τ T τ * a.s. as T ∞, where τ * is defined in (26). Moreover, τ * is optimal in (22).…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bayesian sequential testing of the drift of a Brownian motion

Ekström

Vaicenavicius

2015

ESAIM: PS

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“…However, the optimal boundary is not characterised over the entire time horizon. In a more recent publication by Zhitlukhin and Muravlev (2013), an integral equation is derived for a transformed version of the optimal boundary. This equation can then be solved numerically in order to obtain a full description.…”

Section: Related Literaturementioning

confidence: 99%

Anscombe’s model for sequential clinical trials revisited

Jobjörnsson

Christensen

2018

Sequential Analysis

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Short title (for the running head): Anscombe's Model RevisitedIn Anscombe's classical model, the objective is to find the optimal sequential rule for learning about the difference between two alternative treatments and subsequently selecting the superior one. The population for which the procedure is optimised has size N and includes both the patients in the trial and those which are treated with the chosen alternative after the trial. We review earlier work on this problem and give a detailed treatment of the problem itself. In particular, while previous work has mainly focused on the case of conjugate normal priors for the incremental effect, we demonstrate how to handle the problem for priors of a general form. We also discuss methods for numerical solutions and the practical implications of the results for the regulation of clinical trials.Two extensions of the model are proposed and analysed. The first breaks the symmetry of the treatments, giving one the role of the current standard being administered in parallel with the trial. We show how certain asymptotic results due to Chernoff can be adapted to this asymmetric case. The other extension assumes that N is a random variable instead of a known constant.

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Bayesian and Variational Problems of Hypothesis Testing. Brownian Motion Models

Shiryaev

2019

Stochastic Disorder Problems

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On Chernoff's Hypotheses Testing Problem for the Drift of a Brownian Motion

Cited by 9 publications

References 7 publications

Bayesian sequential testing of the drift of a Brownian motion

Bayesian sequential testing of the drift of a Brownian motion

Anscombe’s model for sequential clinical trials revisited

Bayesian and Variational Problems of Hypothesis Testing. Brownian Motion Models

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