Asymptotic Optimality of Generalized Sequential Likelihood Ratio Tests in Some Classical Sequential Testing Problems*

Lai, T. L.

doi:10.1081/sqa-120016301

Cited by 11 publications

(9 citation statements)

References 32 publications

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“…If G t exceeds a decision limit, h, it is concluded that ξt better represents the current evaluation sample. In usual settings, h is determined based on the asymptotic behavior of the test statistic (e.g., Lai, 1991). In this study, we obtain h via simulation to address the error inherent in the sampling and item calibration.…”

Section: Sglrtmentioning

confidence: 99%

Sequential Generalized Likelihood Ratio Tests for Online Item Monitoring

Kang¹

2021

Preprint

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The study presents statistical procedures that monitor functioning of items over time. We propose generalized likelihood ratio tests that surveil multiple item parameters and implement with various sampling techniques to perform continuous or intermittent monitoring. The procedures examine stability of item parameters across time and inform compromise as soon as they identify significant parameter shift. The performance of the monitoring procedures was validated using simulated and real assessment data. The empirical evaluation suggests that the proposed procedures perform adequately well in identifying the parameter drift. They showed satisfactory detection power and gave timely signals while regulating the error rates reasonably low. The procedures also showed superior performance when compared with the existent methods. The empirical findings suggest that multivariate parametric monitoring can provide an efficient and powerful control tool for maintaining the quality of items. The procedures allow joint monitoring of multiple item parameters and achieve sufficient power by dint of likelihood-ratio tests. Based on the findings from the empirical experimentation, we suggest some practical strategies for performing online item monitoring.

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Section: Sglrtmentioning

confidence: 99%

Sequential Generalized Likelihood Ratio Tests for Online Item Monitoring

Kang¹

2021

Preprint

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“…The main results in the field of sequential hypothesis testing and change-point problems, including the cases of composite hypotheses, were described in the survey papers by Lai (2001Lai ( , 2002. He also investigated the optimality problem for the generalized SPRT (GSPRT).…”

Section: Darkhovskymentioning

confidence: 97%

Optimal Sequential Tests for Testing Two Composite and Multiple Simple Hypotheses

Darkhovsky

2011

Sequential Analysis

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This article considers the problem of sequential testing of two composite hypotheses. Each of the hypotheses is described by a probability density function depending on a parameter. The parameter can belong to one of the two disjoint subsets of a given set. We present a sequential procedure that minimizes the Bayesian risk maximal over a family of prior parameter distributions. The family of prior distributions consists of all of the probabilistic distributions on the parametric set such that the prior probability of one of the hypotheses is equal to a given number. Consider the family of all of the possible sequential decision rules with given constraints on the greatest, with respect to the parameter, error probabilities. We prove that the procedure minimizes the maximal value (with respect to the parameter) of the average run length assuming the validity of any of the above two hypotheses over the family. In the case of two simple hypotheses sequential testing problems, our results give rise to the classical Wald-Wolfowitz theorem. Furthermore, the problem of sequential testing of three, or more, simple hypotheses becomes a special case of our results as well.

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“…For the composite hypothesis testing, Zeitouni, Ziv, and Merhav [8] investigated the generalized likelihood ratio test (GLRT) and proposed conditions for asymptotic optimality of the GLRT in the Neyman-Pearson sense. For the sequential case, Lai [9] analyzed different sequential testing problems and obtained a unified asymptotic theory that results in certain generalized sequential likelihood ratio tests to be asymptotically optimal solutions to these problem. Li, Nitinawarat and Veeravalli [10] considered a universal outlier hypothesis testing problem in the fixed-length setting; universality here refers to the fact that the distributions are unknown and have to be estimated on the fly.…”

Section: A Related Workmentioning

confidence: 99%

Asymptotics of Sequential Composite Hypothesis Testing under Probabilistic Constraints

Pan¹,

Lü²,

Tan³

2021

Preprint

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We consider the sequential composite binary hypothesis testing problem in which one of the hypotheses is governed by a single distribution while the other is governed by a family of distributions whose parameters belong to a known set Γ. We would like to design a test to decide which hypothesis is in effect. Under the constraints that the probabilities that the length of the test, a stopping time, exceeds n are bounded by a certain threshold , we obtain certain fundamental limits on the asymptotic behavior of the sequential test as n tends to infinity. Assuming that Γ is a convex and compact set, we obtain the set of all first-order error exponents for the problem. We also prove a strong converse. Additionally, we obtain the set of second-order error exponents under the assumption that X is a finite alphabet. In the proof of second-order asymptotics, a main technical contribution is the derivation of a central limit-type result for a maximum of an uncountable set of log-likelihood ratios under suitable conditions. This result may be of independent interest. We also show that some important statistical models satisfy the conditions.

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Asymptotic Optimality of Generalized Sequential Likelihood Ratio Tests in Some Classical Sequential Testing Problems*

Cited by 11 publications

References 32 publications

Sequential Generalized Likelihood Ratio Tests for Online Item Monitoring

Sequential Generalized Likelihood Ratio Tests for Online Item Monitoring

Optimal Sequential Tests for Testing Two Composite and Multiple Simple Hypotheses

Asymptotics of Sequential Composite Hypothesis Testing under Probabilistic Constraints

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