Pub Date : 2023-03-21DOI: 10.1080/07474946.2023.2171056
Fatima Ezzahra Mana, Blaise Kévin Guépié, Igor Nikiforov
Abstract This article addresses the sequential detection of transient changes by using the finite moving average (FMA) test. It is assumed that a change occurs at an unknown (but nonrandom) change point and the duration of postchange period is finite and known. We relax the assumption that the profile of a transient change is chosen so that the log-likelihood ratios of the observations are associated random variables (r.v.s) in the prechange mode. Hence, the profile of the transient change is arbitrary and it is not necessarily of constant sign for a distribution with monotone likelihood ratio. A new upper bound for the worst-case probability of false alarm is proposed. It is shown that the optimization of the window-limited cumulative sum (CUSUM) test again leads to the FMA test. Three particular transient changes are considered: in the Gaussian mean, in the Gaussian variance, and in the parameter of exponential distribution. In the first case, a comparison between the bounds for the FMA test operating characteristics and the exact operating characteristics calculated by numerical integration is used to estimate the sharpness of the bounds. In the second and third cases, special attention is paid to the calculation of the FMA distribution in the case of arbitrary profile. The method of convolution is used to solve the problem.
{"title":"Sequential Detection of an Arbitrary Transient Change Profile by the FMA Test","authors":"Fatima Ezzahra Mana, Blaise Kévin Guépié, Igor Nikiforov","doi":"10.1080/07474946.2023.2171056","DOIUrl":"https://doi.org/10.1080/07474946.2023.2171056","url":null,"abstract":"Abstract This article addresses the sequential detection of transient changes by using the finite moving average (FMA) test. It is assumed that a change occurs at an unknown (but nonrandom) change point and the duration of postchange period is finite and known. We relax the assumption that the profile of a transient change is chosen so that the log-likelihood ratios of the observations are associated random variables (r.v.s) in the prechange mode. Hence, the profile of the transient change is arbitrary and it is not necessarily of constant sign for a distribution with monotone likelihood ratio. A new upper bound for the worst-case probability of false alarm is proposed. It is shown that the optimization of the window-limited cumulative sum (CUSUM) test again leads to the FMA test. Three particular transient changes are considered: in the Gaussian mean, in the Gaussian variance, and in the parameter of exponential distribution. In the first case, a comparison between the bounds for the FMA test operating characteristics and the exact operating characteristics calculated by numerical integration is used to estimate the sharpness of the bounds. In the second and third cases, special attention is paid to the calculation of the FMA distribution in the case of arbitrary profile. The method of convolution is used to solve the problem.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46063624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-02DOI: 10.1080/07474946.2022.2150778
C. Makasu
Abstract In this paper, we establish a one-sided maximal moment inequality with exact constants for Bessel processes. As a consequence, we obtain an exact constant in the Burkholder-Gundy inequality. The proof of our main result is based on a pure optimal stopping problem of the running maximum process for a Bessel process. The present results extend and complement a number of related results previously known in the literature.
{"title":"On the exact constants in one-sided maximal inequalities for Bessel processes","authors":"C. Makasu","doi":"10.1080/07474946.2022.2150778","DOIUrl":"https://doi.org/10.1080/07474946.2022.2150778","url":null,"abstract":"Abstract In this paper, we establish a one-sided maximal moment inequality with exact constants for Bessel processes. As a consequence, we obtain an exact constant in the Burkholder-Gundy inequality. The proof of our main result is based on a pure optimal stopping problem of the running maximum process for a Bessel process. The present results extend and complement a number of related results previously known in the literature.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42107766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-02DOI: 10.1080/07474946.2022.2154364
Leng-Cheng Hwang
Abstract Within the Bayesian framework, a robust two-stage procedure is proposed to deal with the problem of multivariate sequential estimation of the unknown mean vector with weighted squared error loss and fixed cost per observation. The proposed procedure depends on the present data but not on the distributions of outcome variables or the prior. It is shown that the proposed procedure shares the asymptotic properties with the optimal fixed-sample-size procedures for the arbitrary distributions and the asymptotically pointwise optimal procedures for the distributions of a multivariate exponential family with a large class of prior distributions. Simulation results indicate that the proposed two-stage procedure is robust to misspecification of the true parameters of the prior distribution and outperforms the purely sequential procedure and the asymptotically pointwise optimal procedure in terms of robustness.
{"title":"Asymptotic optimality of a robust two-stage procedure in multivariate Bayes sequential estimation","authors":"Leng-Cheng Hwang","doi":"10.1080/07474946.2022.2154364","DOIUrl":"https://doi.org/10.1080/07474946.2022.2154364","url":null,"abstract":"Abstract Within the Bayesian framework, a robust two-stage procedure is proposed to deal with the problem of multivariate sequential estimation of the unknown mean vector with weighted squared error loss and fixed cost per observation. The proposed procedure depends on the present data but not on the distributions of outcome variables or the prior. It is shown that the proposed procedure shares the asymptotic properties with the optimal fixed-sample-size procedures for the arbitrary distributions and the asymptotically pointwise optimal procedures for the distributions of a multivariate exponential family with a large class of prior distributions. Simulation results indicate that the proposed two-stage procedure is robust to misspecification of the true parameters of the prior distribution and outperforms the purely sequential procedure and the asymptotically pointwise optimal procedure in terms of robustness.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44434267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-02DOI: 10.1080/07474946.2022.2159042
Elena M. Buzaianu, Pinyuen Chen, Lifang Hsu
Abstract This article considers the goal of selecting the population with the largest mean among k normal populations when variances are not known. We propose a Stein-type two-sample procedure, denoted by for selecting a nonempty random-size subset of size at most m ( ) that contains the population associated with the largest mean, with a guaranteed minimum probability whenever the distance between the largest mean and the second largest mean is at least where m, and are specified in advance of the experiment. The probability of a correct selection and the expected subset size of are derived. Critical values/procedure parameters that are required for certain k, m, and are obtained by solving simultaneous integral equations and are presented in tables.
{"title":"A restricted subset selection procedure for selecting the largest normal mean under heteroscedasticity","authors":"Elena M. Buzaianu, Pinyuen Chen, Lifang Hsu","doi":"10.1080/07474946.2022.2159042","DOIUrl":"https://doi.org/10.1080/07474946.2022.2159042","url":null,"abstract":"Abstract This article considers the goal of selecting the population with the largest mean among k normal populations when variances are not known. We propose a Stein-type two-sample procedure, denoted by for selecting a nonempty random-size subset of size at most m ( ) that contains the population associated with the largest mean, with a guaranteed minimum probability whenever the distance between the largest mean and the second largest mean is at least where m, and are specified in advance of the experiment. The probability of a correct selection and the expected subset size of are derived. Critical values/procedure parameters that are required for certain k, m, and are obtained by solving simultaneous integral equations and are presented in tables.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41911034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-02DOI: 10.1080/07474946.2022.2123517
P. Sultana, M. Pal, B. Sinha
Abstract Sequential sampling plans for unbiased estimation of the Bernoulli parameter ‘p’ and its functions have been studied almost 70 years back. An extension of the idea to parametric function estimation in a tetranomial distribution has been considered in this paper. The cell probabilities are taken to be p 2, q 2, r 2 and 2(pq+pr+qr), satisfying p, q, r > 0, p +q +r = 1. Some illustrative examples have been studied to demonstrate the underlying concepts and the computational procedure. The emphasis is on unbiased estimation in the exact sense which makes it difficult and challenging.
{"title":"Exact Inference in a Tetranomial Distribution","authors":"P. Sultana, M. Pal, B. Sinha","doi":"10.1080/07474946.2022.2123517","DOIUrl":"https://doi.org/10.1080/07474946.2022.2123517","url":null,"abstract":"Abstract Sequential sampling plans for unbiased estimation of the Bernoulli parameter ‘p’ and its functions have been studied almost 70 years back. An extension of the idea to parametric function estimation in a tetranomial distribution has been considered in this paper. The cell probabilities are taken to be p 2, q 2, r 2 and 2(pq+pr+qr), satisfying p, q, r > 0, p +q +r = 1. Some illustrative examples have been studied to demonstrate the underlying concepts and the computational procedure. The emphasis is on unbiased estimation in the exact sense which makes it difficult and challenging.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48605749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01Epub Date: 2023-05-23DOI: 10.1080/07474946.2023.2187417
Qunzhi Xu, Yajun Mei
The active quickest detection problem with unknown post-change parameters is studied under the sampling control constraint, where there are p local streams in a system but one is only able to take observations from one and only one of these p local streams at each time instant. The objective is to raise a correct alarm as quickly as possible once the change occurs subject to both false alarm and sampling control constraints. Here we assume that exactly one of the p local streams is affected, and the post-change distribution involves unknown parameters. In this context, we propose an efficient greedy-cyclic-sampling-based quickest detection algorithm, and show that our proposed algorithm is asymptotically optimal in the sense of minimizing the detection delay under both false alarm and sampling control constraints. Numerical studies are conducted to show the effectiveness and applicability of the proposed algorithm.
在采样控制约束条件下，研究了具有未知变化后参数的主动最快检测问题，即系统中有 p 个本地流，但在每个时间瞬时只能从这 p 个本地流中的一个进行观测。我们的目标是，一旦发生变化，在误报和采样控制的约束下，尽快发出正确的警报。在此，我们假设 p 个本地流中正好有一个受到影响，而变化后的分布涉及未知参数。在这种情况下，我们提出了一种高效的基于贪婪循环采样的最快检测算法，并证明了我们提出的算法在误报和采样控制约束条件下检测延迟最小的意义上是渐近最优的。我们还进行了数值研究，以证明所提算法的有效性和适用性。
{"title":"Asymptotic Optimality Theory for Active Quickest Detection with Unknown PostChange Parameters.","authors":"Qunzhi Xu, Yajun Mei","doi":"10.1080/07474946.2023.2187417","DOIUrl":"10.1080/07474946.2023.2187417","url":null,"abstract":"<p><p>The active quickest detection problem with unknown post-change parameters is studied under the sampling control constraint, where there are <i>p</i> local streams in a system but one is only able to take observations from one and only one of these <i>p</i> local streams at each time instant. The objective is to raise a correct alarm as quickly as possible once the change occurs subject to both false alarm and sampling control constraints. Here we assume that exactly one of the <i>p</i> local streams is affected, and the post-change distribution involves unknown parameters. In this context, we propose an efficient greedy-cyclic-sampling-based quickest detection algorithm, and show that our proposed algorithm is asymptotically optimal in the sense of minimizing the detection delay under both false alarm and sampling control constraints. Numerical studies are conducted to show the effectiveness and applicability of the proposed algorithm.</p>","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10462384/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10500881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-21DOI: 10.1080/07474946.2022.2149804
P. Jeyadurga, S. Balamurali
Abstract Designing of acceptance sampling plans based on operating characteristics fail to prescribe the corrective action for rejected lots although they guarantee the producer’s and the consumer’s protection at their corresponding quality levels. But, rectification sampling inspection plan suggests 100% inspection for rejected lots and hence, the producer’s risk is reduced and also there is a guarantee to the consumer by restricting upper limit of the maximum expected proportion of non-conforming of the shipping lot. Although the variables single sampling plan for rectifying inspection is available, variables double sampling inspection plan under rectifying inspection has not yet been designed. Hence, in this paper, we design a variables double sampling plan for rectifying inspection indexed by acceptable quality level and average outgoing quality limit to meet the needs of the producer as well as the consumer. A non-linear optimization problem is developed to determine the optimal parameters with minimum average sample number at acceptable quality level for both cases of standard deviation are known and unknown. We investigate the validation and limitation of the proposed plan through the simulation study and industrial application of the proposed plan is also given. It can be confirmed from the investigations on the results that in many situations, variables double sampling plan designed under rectifying inspection is more economical than variables single sampling plan designed with the same conditions in providing the same protection to consumer.
{"title":"Designing of Variables Double Sampling Plan Under Rectifying Inspection for Consumer Protection","authors":"P. Jeyadurga, S. Balamurali","doi":"10.1080/07474946.2022.2149804","DOIUrl":"https://doi.org/10.1080/07474946.2022.2149804","url":null,"abstract":"Abstract Designing of acceptance sampling plans based on operating characteristics fail to prescribe the corrective action for rejected lots although they guarantee the producer’s and the consumer’s protection at their corresponding quality levels. But, rectification sampling inspection plan suggests 100% inspection for rejected lots and hence, the producer’s risk is reduced and also there is a guarantee to the consumer by restricting upper limit of the maximum expected proportion of non-conforming of the shipping lot. Although the variables single sampling plan for rectifying inspection is available, variables double sampling inspection plan under rectifying inspection has not yet been designed. Hence, in this paper, we design a variables double sampling plan for rectifying inspection indexed by acceptable quality level and average outgoing quality limit to meet the needs of the producer as well as the consumer. A non-linear optimization problem is developed to determine the optimal parameters with minimum average sample number at acceptable quality level for both cases of standard deviation are known and unknown. We investigate the validation and limitation of the proposed plan through the simulation study and industrial application of the proposed plan is also given. It can be confirmed from the investigations on the results that in many situations, variables double sampling plan designed under rectifying inspection is more economical than variables single sampling plan designed with the same conditions in providing the same protection to consumer.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47704230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-10DOI: 10.1080/07474946.2023.2215825
A. Novikov
Abstract In this article, we deal with the problem of sequential testing of multiple hypotheses. The main goal is minimizing the expected sample size (ESS) under restrictions on the error probabilities. We take, as a criterion of minimization, a weighted sum of the ESSs evaluated at some points of interest in the parameter space, aiming at its minimization under restrictions on the error probabilities. We use a variant of the method of Lagrange multipliers based on the minimization of an auxiliary objective function (called Lagrangian) combining the objective function with the restrictions, taken with some constants called multipliers. Subsequently, the multipliers are used to make the solution comply with the restrictions. We develop a computer-oriented method of minimization of the Lagrangian function that provides, depending on the specific choice of the parameter points, optimal tests in different concrete settings, as in Bayesian, Kiefer-Weiss, and other settings. To exemplify the proposed methods for the particular case of sampling from a Bernoulli population, we develop a set of computer algorithms for designing sequential tests that minimize the Lagrangian function and for the numerical evaluation of test characteristics like the error probabilities and the ESS. We implement the algorithms in the R programming language. The program code is available in a public GitHub repository. For the Bernoulli model, we made a series of computer evaluations related to the optimality of sequential multi-hypothesis tests, in a particular case of three hypotheses. A numerical comparison with the matrix sequential probability ratio test is carried out. A method of solution of the multi-hypothesis Kiefer-Weiss problem is proposed and is applied for a particular case of three hypotheses in the Bernoulli model.
{"title":"A numerical approach to sequential multi-hypothesis testing for Bernoulli model","authors":"A. Novikov","doi":"10.1080/07474946.2023.2215825","DOIUrl":"https://doi.org/10.1080/07474946.2023.2215825","url":null,"abstract":"Abstract In this article, we deal with the problem of sequential testing of multiple hypotheses. The main goal is minimizing the expected sample size (ESS) under restrictions on the error probabilities. We take, as a criterion of minimization, a weighted sum of the ESSs evaluated at some points of interest in the parameter space, aiming at its minimization under restrictions on the error probabilities. We use a variant of the method of Lagrange multipliers based on the minimization of an auxiliary objective function (called Lagrangian) combining the objective function with the restrictions, taken with some constants called multipliers. Subsequently, the multipliers are used to make the solution comply with the restrictions. We develop a computer-oriented method of minimization of the Lagrangian function that provides, depending on the specific choice of the parameter points, optimal tests in different concrete settings, as in Bayesian, Kiefer-Weiss, and other settings. To exemplify the proposed methods for the particular case of sampling from a Bernoulli population, we develop a set of computer algorithms for designing sequential tests that minimize the Lagrangian function and for the numerical evaluation of test characteristics like the error probabilities and the ESS. We implement the algorithms in the R programming language. The program code is available in a public GitHub repository. For the Bernoulli model, we made a series of computer evaluations related to the optimality of sequential multi-hypothesis tests, in a particular case of three hypotheses. A numerical comparison with the matrix sequential probability ratio test is carried out. A method of solution of the multi-hypothesis Kiefer-Weiss problem is proposed and is applied for a particular case of three hypotheses in the Bernoulli model.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47765666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-31DOI: 10.1080/07474946.2023.2211126
Grigory Sokolov, V. Spivak, A. Tartakovsky
Abstract Oftentimes, in practice, the observed process changes statistical properties at an unknown point in time and the duration of a change is substantially finite, in which case one says that the change is intermittent or transient. We provide an overview of existing approaches for intermittent change detection and advocate in favor of a particular setting driven by the intermittent nature of the change. We propose a novel optimization criterion that is more appropriate for many applied areas such as the detection of threats in physical computer systems, near-Earth space informatics, epidemiology, pharmacokinetics, etc. We argue that controlling the local conditional probability of a false alarm, rather than the familiar average run length to a false alarm, and maximizing the local conditional probability of detection is a more reasonable approach versus a traditional quickest change detection approach that requires minimizing the expected delay to detection. We adopt the maximum likelihood (ML) approach with respect to the change duration and show that several commonly used detection rules (cumulative sum [CUSUM], window-limited [WL]-CUSUM, and finite moving average [FMA]) are equivalent to the ML-based stopping times. We discuss how to choose design parameters for these rules and provide a comprehensive simulation study to corroborate intuitive expectations.
{"title":"Detecting an intermittent change of unknown duration","authors":"Grigory Sokolov, V. Spivak, A. Tartakovsky","doi":"10.1080/07474946.2023.2211126","DOIUrl":"https://doi.org/10.1080/07474946.2023.2211126","url":null,"abstract":"Abstract Oftentimes, in practice, the observed process changes statistical properties at an unknown point in time and the duration of a change is substantially finite, in which case one says that the change is intermittent or transient. We provide an overview of existing approaches for intermittent change detection and advocate in favor of a particular setting driven by the intermittent nature of the change. We propose a novel optimization criterion that is more appropriate for many applied areas such as the detection of threats in physical computer systems, near-Earth space informatics, epidemiology, pharmacokinetics, etc. We argue that controlling the local conditional probability of a false alarm, rather than the familiar average run length to a false alarm, and maximizing the local conditional probability of detection is a more reasonable approach versus a traditional quickest change detection approach that requires minimizing the expected delay to detection. We adopt the maximum likelihood (ML) approach with respect to the change duration and show that several commonly used detection rules (cumulative sum [CUSUM], window-limited [WL]-CUSUM, and finite moving average [FMA]) are equivalent to the ML-based stopping times. We discuss how to choose design parameters for these rules and provide a comprehensive simulation study to corroborate intuitive expectations.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43411668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-02DOI: 10.1080/07474946.2022.2129689
B. Levin, C. Leu
Abstract Levin and Leu (2021) introduced some key inequalities that underlie the lower bound formula for the probability of lattice events when using adaptive members of the Levin-Robbins-Leu family of sequential subset selection procedures for binary outcomes. Here we provide a rigorous proof of the key inequality for each adaptive procedure in the special case of equal odds parameters. We also provide some further insight into why the key inequality holds for arbitrary odds parameters and we present a complete proof in that case for a simple yet non-trivial prototype example. Two errata in the abovementioned publication are also corrected herein.
{"title":"Proof of a Key Inequality for Lattice Event Probabilities with Equal Odds","authors":"B. Levin, C. Leu","doi":"10.1080/07474946.2022.2129689","DOIUrl":"https://doi.org/10.1080/07474946.2022.2129689","url":null,"abstract":"Abstract Levin and Leu (2021) introduced some key inequalities that underlie the lower bound formula for the probability of lattice events when using adaptive members of the Levin-Robbins-Leu family of sequential subset selection procedures for binary outcomes. Here we provide a rigorous proof of the key inequality for each adaptive procedure in the special case of equal odds parameters. We also provide some further insight into why the key inequality holds for arbitrary odds parameters and we present a complete proof in that case for a simple yet non-trivial prototype example. Two errata in the abovementioned publication are also corrected herein.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47986061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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