Optimal Sequential Tests for Two Simple Hypotheses

Novikov, Andrey

doi:10.1080/07474940902816809

Cited by 41 publications

(87 citation statements)

References 13 publications

(6 reference statements)

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“…The proof of Lemma 1 is almost identical to that of Lemma 3.5 in [27]. Below in this section, under more specific conditions, we give less trivial (and more interesting from the theoretical and practical points of view) examples of F satisfying Assumption 1.…”

Section: General Stopping Rulesmentioning

confidence: 89%

“…It is worth noting that the method developed in Section 2 has been applied (for particular processes) to solving several general problems in statistical sequential analysis (see [26][27][28]). Further possible applications of our results could appear in models of statistical sequential analysis with dependent observations, in detecting changes in non-Markovian discrete-time processes, and in some models of risk theory.…”

Section: Introductionmentioning

confidence: 99%

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Characterizations of Optimal Policies in a General Stopping Problem and Stability Estimating

Gordienko¹,

Novikov²

2014

Prob. Eng. Inf. Sci.

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We consider an optimal stopping problem for a general discrete-time process X 1 , X 2 , . . . , Xn, . . . on a common measurable space. Stopping at time n (n = 1, 2, . . . ) yields a reward Rn(X 1 , . . . , Xn) ≥ 0, while if we do not stop, we pay cn(X 1 , . . . , Xn) ≥ 0 and keep observing the process. The problem is to characterize all the optimal stopping times τ , i.e., such that maximize the mean net gain:cn(X 1 , . . . , Xn)).We propose a new simple approach to stopping problems which allows to obtain not only sufficient, but also necessary conditions of optimality in some natural classes of (randomized) stopping rules.In the particular case of Markov sequence X 1 , X 2 , . . . we estimate the stability of the optimal stopping problem under perturbations of transition probabilities.

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Section: General Stopping Rulesmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Characterizations of Optimal Policies in a General Stopping Problem and Stability Estimating

Gordienko¹,

Novikov²

2014

Prob. Eng. Inf. Sci.

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“…( 3.17) The optimal stopping rule Ψ n follows directly from the definition of ρ n , see, e.g., (Novikov, 2009;Poor and Hadjiliadis, 2009).…”

Section: )mentioning

confidence: 99%

“…The proof of Theorem 3.1 follows from the fundamental results of optimal stopping theory. We follow here the proofs stated in Fauß and Zoubir (2015); Novikov (2009);Poor and Hadjiliadis (2009). In this proof, a truncated optimal stopping problem with finite horizon N ≥ 1 is considered with the cost c n for stopping at time n. Let V n = min{c n , E [V n+1 | t n ]} , n < N be the cost for stopping at the optimal time instant between n and N with basis V N = c N .…”

Section: A Proof Of Theorem 31mentioning

confidence: 99%

Bayesian sequential joint detection and estimation

Reinhard

Fauß

Zoubir

2018

Sequential Analysis

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Joint detection and estimation refers to deciding between two or more hypotheses and, depending on the test outcome, simultaneously estimating the unknown parameters of the underlying distribution. This problem is investigated in a sequential framework under mild assumptions on the underlying random process. We formulate an unconstrained sequential decision problem, whose cost function is the weighted sum of the expected run-length and the detection/estimation errors. Then, a strong connection between the derivatives of the cost function with respect to the weights, which can be interpreted as Lagrange multipliers, and the detection/estimation errors of the underlying scheme is shown. This property is used to characterize the solution of a closely related sequential decision problem, whose objective function is the expected run-length under constraints on the average detection/estimation errors. We show that the solution of the constrained problem coincides with the solution of the unconstrained problem with suitably chosen weights. These weights are characterized as the solution of a linear program, which can be solved using efficient off-the-shelf solvers. The theoretical results are illustrated with two example problems, for which optimal sequential schemes are designed numerically and whose performance is validated via Monte Carlo simulations.2. The hypothesis H i , i = 0, 1, as well as the random parameter Θ i do not change during the observation period of X N .3. A sufficient static t n (x n ) in a state space (E t , E t ) exists such thatThe sufficient statistic has a transition kernel of the formand some initial statistic t 0 .4. The two hypotheses are separable, i.e.,5. The second order moment of the random parameter Θ exists and is finite, i.e., E[Θ 2 ] <∞ .Note that this implies, that the conditional second order moment E[Θ 2 | A] exist and is finite for all events A with non-zero probability.

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“…where ρ k ∈ P. Following the techniques developed in [22], [23], (14) can be straightforwardly solved as follows, where we omit the details in the interest of space: Let m0 and m1 denote the number of 0's and 1's observed. The likelihood-ratios of the corresponding observations under P0 and Pρ k , with respect to Pρ * are given by…”

Section: Joint Detection and Estimationmentioning

confidence: 99%

Sequential joint signal detection and signal-to-noise ratio estimation

Fauß

Nagananda

Zoubir

et al. 2017

2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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The sequential analysis of the problem of joint signal detection and signal-to-noise ratio (SNR) estimation for a linear Gaussian observation model is considered. The problem is posed as an optimization setup where the goal is to minimize the number of samples required to achieve the desired (i) type I and type II error probabilities and (ii) mean squared error performance. This optimization problem is reduced to a more tractable formulation by transforming the observed signal and noise sequences to a single sequence of Bernoulli random variables; joint detection and estimation is then performed on the Bernoulli sequence. This transformation renders the problem easily solvable, and results in a computationally simpler sufficient statistic compared to the one based on the (untransformed) observation sequences. Experimental results demonstrate the advantages of the proposed method, making it feasible for applications having strict constraints on data storage and computation.

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Optimal Sequential Tests for Two Simple Hypotheses

Cited by 41 publications

References 13 publications

Characterizations of Optimal Policies in a General Stopping Problem and Stability Estimating

Characterizations of Optimal Policies in a General Stopping Problem and Stability Estimating

Bayesian sequential joint detection and estimation

Sequential joint signal detection and signal-to-noise ratio estimation

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