Pub Date : 2020-01-15DOI: 10.1080/07474946.2020.1823192
Nils Mause, A. Steland
Abstract The K sample problem for high-dimensional vector time series is studied, especially focusing on sensor data streams, in order to analyze the second moment structure and detect changes across samples and/or across variables cumulated sum (CUSUM) statistics of bilinear forms of the sample covariance matrix. In this model, K independent vector time series are observed over a time span which may correspond to K sensors (locations) yielding d-dimensional data as well as K locations where d sensors emit univariate data. Unequal sample sizes are considered as arising when the sampling rate of the sensors differs. We provide large-sample approximations and two related change point statistics, a sum of squares and a pooled variance statistic. The resulting procedures are investigated by simulations and illustrated by analyzing a real data set.
{"title":"Detecting changes in the second moment structure of high-dimensional sensor-type data in a K-sample setting","authors":"Nils Mause, A. Steland","doi":"10.1080/07474946.2020.1823192","DOIUrl":"https://doi.org/10.1080/07474946.2020.1823192","url":null,"abstract":"Abstract The K sample problem for high-dimensional vector time series is studied, especially focusing on sensor data streams, in order to analyze the second moment structure and detect changes across samples and/or across variables cumulated sum (CUSUM) statistics of bilinear forms of the sample covariance matrix. In this model, K independent vector time series are observed over a time span which may correspond to K sensors (locations) yielding d-dimensional data as well as K locations where d sensors emit univariate data. Unequal sample sizes are considered as arising when the sampling rate of the sensors differs. We provide large-sample approximations and two related change point statistics, a sum of squares and a pooled variance statistic. The resulting procedures are investigated by simulations and illustrated by analyzing a real data set.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":"39 1","pages":"336 - 366"},"PeriodicalIF":0.8,"publicationDate":"2020-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2020.1823192","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49194603","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 : 2020-01-02DOI: 10.1080/07474946.2020.1726678
Georgios Rovatsos, Shaofeng Zou, V. Veeravalli
Abstract The problem of quickest moving anomaly detection in networks is studied. Initially, the observations are generated according to a prechange distribution. At some unknown but deterministic time, an anomaly emerges in the network. At each time instant, one node is affected by the anomaly and receives data from a post-change distribution. The anomaly moves across the network, and the node that it affects changes with time. However, the trajectory of the moving anomaly is assumed to be unknown. A discrete-time Markov chain is employed to model the unknown trajectory of the moving anomaly in the network. A windowed generalized likelihood ratio–based test is constructed and is shown to be asymptotically optimal. Other detection algorithms including the dynamic Shiryaev-Roberts test, a quickest change detection algorithm with recursive change point estimation, and a mixture cumulative sum (CUSUM) algorithm are also developed for this problem. Lower bounds on the mean time to false alarm are developed. Numerical results are further provided to compare their performances.
{"title":"Sequential algorithms for moving anomaly detection in networks","authors":"Georgios Rovatsos, Shaofeng Zou, V. Veeravalli","doi":"10.1080/07474946.2020.1726678","DOIUrl":"https://doi.org/10.1080/07474946.2020.1726678","url":null,"abstract":"Abstract The problem of quickest moving anomaly detection in networks is studied. Initially, the observations are generated according to a prechange distribution. At some unknown but deterministic time, an anomaly emerges in the network. At each time instant, one node is affected by the anomaly and receives data from a post-change distribution. The anomaly moves across the network, and the node that it affects changes with time. However, the trajectory of the moving anomaly is assumed to be unknown. A discrete-time Markov chain is employed to model the unknown trajectory of the moving anomaly in the network. A windowed generalized likelihood ratio–based test is constructed and is shown to be asymptotically optimal. Other detection algorithms including the dynamic Shiryaev-Roberts test, a quickest change detection algorithm with recursive change point estimation, and a mixture cumulative sum (CUSUM) algorithm are also developed for this problem. Lower bounds on the mean time to false alarm are developed. Numerical results are further provided to compare their performances.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":"39 1","pages":"31 - 6"},"PeriodicalIF":0.8,"publicationDate":"2020-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2020.1726678","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47495857","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 : 2020-01-02DOI: 10.1080/07474946.2020.1727166
K. Kachiashvili, J. K. Kachiashvili, I. Prangishvili
Abstract Constrained Bayesian methods (CBMs) and the concept of false discovery rates (FDRs) for testing directional hypotheses are considered in this article. It is shown that the direct application of CBM allows us to control FDR on the desired level for both one set of directional hypotheses and a multiple case when we consider () sets of directional hypotheses. When guaranteeing restriction on the desired level, a Bayesian sequential method can be applied, the stopping rules of which are proper and the sequential scheme for making a decision strongly controls the mixed directional FDR. Computational results of concrete examples confirm the correctness of the theoretical outcomes.
{"title":"CBM for testing multiple hypotheses with directional alternatives in sequential experiments","authors":"K. Kachiashvili, J. K. Kachiashvili, I. Prangishvili","doi":"10.1080/07474946.2020.1727166","DOIUrl":"https://doi.org/10.1080/07474946.2020.1727166","url":null,"abstract":"Abstract Constrained Bayesian methods (CBMs) and the concept of false discovery rates (FDRs) for testing directional hypotheses are considered in this article. It is shown that the direct application of CBM allows us to control FDR on the desired level for both one set of directional hypotheses and a multiple case when we consider () sets of directional hypotheses. When guaranteeing restriction on the desired level, a Bayesian sequential method can be applied, the stopping rules of which are proper and the sequential scheme for making a decision strongly controls the mixed directional FDR. Computational results of concrete examples confirm the correctness of the theoretical outcomes.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":"39 1","pages":"115 - 131"},"PeriodicalIF":0.8,"publicationDate":"2020-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2020.1727166","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47089721","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 : 2020-01-02DOI: 10.1080/07474946.2020.1726681
Chahira Allouti, Bahia Barache, A. Dahmani
Abstract In this article, we investigate the problem of approximating the fixed point for some function using a Mann iterative process with random errors. After establishing some exponential inequalities, we prove the complete convergence of Mann’s algorithm toward the fixed point and deduce a confidence interval for this one. In addition, we establish the convergence rate of Mann’s algorithm. Several numerical examples are sketched to illustrate the performance of the proposed algorithm.
{"title":"Exponential inequalities for Mann’s stochastic algorithm","authors":"Chahira Allouti, Bahia Barache, A. Dahmani","doi":"10.1080/07474946.2020.1726681","DOIUrl":"https://doi.org/10.1080/07474946.2020.1726681","url":null,"abstract":"Abstract In this article, we investigate the problem of approximating the fixed point for some function using a Mann iterative process with random errors. After establishing some exponential inequalities, we prove the complete convergence of Mann’s algorithm toward the fixed point and deduce a confidence interval for this one. In addition, we establish the convergence rate of Mann’s algorithm. Several numerical examples are sketched to illustrate the performance of the proposed algorithm.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":"39 1","pages":"32 - 51"},"PeriodicalIF":0.8,"publicationDate":"2020-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2020.1726681","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45980441","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 : 2019-10-24DOI: 10.1080/07474946.2021.1912525
Aditya Deshmukh, S. Bhashyam, V. Veeravalli
Abstract The problem of multihypothesis testing with controlled sensing of observations is considered. The distribution of observations collected under each control is assumed to follow a single-parameter exponential family distribution. The goal is to design a policy to find the true hypothesis with minimum expected delay while ensuring that the probability of error is below a given constraint. The decision-maker can reduce the delay by intelligently choosing the control for observation collection in each time slot. A policy for this problem is derived that satisfies given constraints on the error probability, and it is shown that this policy is asymptotically optimal in the sense that it asymptotically achieves an information-theoretic lower bound on the expected delay. Numerical results are provided that illustrate an application of the policy to medical diagnostic inference.
{"title":"Sequential controlled sensing for composite multihypothesis testing","authors":"Aditya Deshmukh, S. Bhashyam, V. Veeravalli","doi":"10.1080/07474946.2021.1912525","DOIUrl":"https://doi.org/10.1080/07474946.2021.1912525","url":null,"abstract":"Abstract The problem of multihypothesis testing with controlled sensing of observations is considered. The distribution of observations collected under each control is assumed to follow a single-parameter exponential family distribution. The goal is to design a policy to find the true hypothesis with minimum expected delay while ensuring that the probability of error is below a given constraint. The decision-maker can reduce the delay by intelligently choosing the control for observation collection in each time slot. A policy for this problem is derived that satisfies given constraints on the error probability, and it is shown that this policy is asymptotically optimal in the sense that it asymptotically achieves an information-theoretic lower bound on the expected delay. Numerical results are provided that illustrate an application of the policy to medical diagnostic inference.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":"40 1","pages":"259 - 289"},"PeriodicalIF":0.8,"publicationDate":"2019-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2021.1912525","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47409414","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 : 2019-10-07DOI: 10.1080/07474946.2021.1847966
A. Steland, Yuan-Tsung Chang
Abstract Nonparametric two-stage procedures to construct fixed-width confidence intervals are studied to quantify uncertainty. It is shown that the validity of the random central limit theorem (RCLT) accompanied by a consistent and asymptotically unbiased estimator of the asymptotic variance already guarantees consistency and first-order as well as second-order efficiency of the two-stage procedures. This holds under the common asymptotics where the length of the confidence interval tends toward 0 as well as under the novel proposed high-confidence asymptotics where the confidence level tends toward 1. The approach is motivated by and applicable to data analysis from distributed big data with nonnegligible costs of data queries. The following problems are discussed: Fixed-width intervals for the mean, for a projection when observing high-dimensional data, and for the common mean when using nonlinear common mean estimators under order constraints. The procedures are investigated by simulations and illustrated by a real data analysis.
{"title":"High-confidence nonparametric fixed-width uncertainty intervals and applications to projected high-dimensional data and common mean estimation","authors":"A. Steland, Yuan-Tsung Chang","doi":"10.1080/07474946.2021.1847966","DOIUrl":"https://doi.org/10.1080/07474946.2021.1847966","url":null,"abstract":"Abstract Nonparametric two-stage procedures to construct fixed-width confidence intervals are studied to quantify uncertainty. It is shown that the validity of the random central limit theorem (RCLT) accompanied by a consistent and asymptotically unbiased estimator of the asymptotic variance already guarantees consistency and first-order as well as second-order efficiency of the two-stage procedures. This holds under the common asymptotics where the length of the confidence interval tends toward 0 as well as under the novel proposed high-confidence asymptotics where the confidence level tends toward 1. The approach is motivated by and applicable to data analysis from distributed big data with nonnegligible costs of data queries. The following problems are discussed: Fixed-width intervals for the mean, for a projection when observing high-dimensional data, and for the common mean when using nonlinear common mean estimators under order constraints. The procedures are investigated by simulations and illustrated by a real data analysis.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":"40 1","pages":"97 - 124"},"PeriodicalIF":0.8,"publicationDate":"2019-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2021.1847966","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47363101","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 : 2019-10-02DOI: 10.1080/07474946.2019.1686937
N. Mukhopadhyay, Soumik Banerjee
Abstract We have designed a sequential minimum risk point estimation (MRPE) strategy for the unknown mean of a normal population having its variance unknown too. This is developed under a Linex loss plus linear cost of sampling. A number of important asymptotic first-order and asymptotic second-order properties' characteristics have been developed and proved thoroughly. Extensive sets of simulations tend to validate nearly all of these asymptotic properties for small to medium to large optimal fixed sample sizes.
{"title":"Sequential minimum risk point estimation (MRPE) methodology for a normal mean under Linex loss plus sampling cost: First-order and second-order asymptotics","authors":"N. Mukhopadhyay, Soumik Banerjee","doi":"10.1080/07474946.2019.1686937","DOIUrl":"https://doi.org/10.1080/07474946.2019.1686937","url":null,"abstract":"Abstract We have designed a sequential minimum risk point estimation (MRPE) strategy for the unknown mean of a normal population having its variance unknown too. This is developed under a Linex loss plus linear cost of sampling. A number of important asymptotic first-order and asymptotic second-order properties' characteristics have been developed and proved thoroughly. Extensive sets of simulations tend to validate nearly all of these asymptotic properties for small to medium to large optimal fixed sample sizes.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":"38 1","pages":"461 - 479"},"PeriodicalIF":0.8,"publicationDate":"2019-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2019.1686937","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46212633","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 : 2019-10-02DOI: 10.1080/07474946.2019.1686938
{"title":"Thanks to the Referees","authors":"","doi":"10.1080/07474946.2019.1686938","DOIUrl":"https://doi.org/10.1080/07474946.2019.1686938","url":null,"abstract":"","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":"38 1","pages":"569 - 569"},"PeriodicalIF":0.8,"publicationDate":"2019-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2019.1686938","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46831988","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 : 2019-10-02DOI: 10.1080/07474946.2019.1686885
N. Mukhopadhyay, Zhe Wang
Abstract A purely sequential minimum risk point estimation (MRPE) methodology with associated stopping time N is designed to come up with a useful estimation strategy. We work under an appropriately formulated weighted squared error loss (SEL) due to estimation of a function of μ, with plus linear cost of sampling from a population having both parameters unknown. A series of important first-order and second-order asymptotic (as c, the cost per unit sample, ) results is laid out including the first-order and second-order efficiency properties. Then, accurate sequential risk calculations are launched, which are then followed by two main results: (i) Theorem 4.1 shows an asymptotic risk efficiency property, and (ii) Theorem 5.1 shows an asymptotic second-order regret expansion associated with the proposed purely sequential MRPE strategy assuming suitable conditions on g(.). We also provide a bias-corrected version of the terminal estimator, We follow up with a number of interesting illustrations where Theorems 4.1–5.1 are readily exploited to conclude an asymptotic risk efficiency property and second-order regret expansion, respectively. A number of other interesting illustrations are highlighted where it is possible to verify the conclusions from Theorems 4.1–5.1 more directly with less stringent assumptions on the pilot sample size.
{"title":"A general theory of purely sequential minimum risk point estimation (MRPE) of a function of the mean in a normal distribution","authors":"N. Mukhopadhyay, Zhe Wang","doi":"10.1080/07474946.2019.1686885","DOIUrl":"https://doi.org/10.1080/07474946.2019.1686885","url":null,"abstract":"Abstract A purely sequential minimum risk point estimation (MRPE) methodology with associated stopping time N is designed to come up with a useful estimation strategy. We work under an appropriately formulated weighted squared error loss (SEL) due to estimation of a function of μ, with plus linear cost of sampling from a population having both parameters unknown. A series of important first-order and second-order asymptotic (as c, the cost per unit sample, ) results is laid out including the first-order and second-order efficiency properties. Then, accurate sequential risk calculations are launched, which are then followed by two main results: (i) Theorem 4.1 shows an asymptotic risk efficiency property, and (ii) Theorem 5.1 shows an asymptotic second-order regret expansion associated with the proposed purely sequential MRPE strategy assuming suitable conditions on g(.). We also provide a bias-corrected version of the terminal estimator, We follow up with a number of interesting illustrations where Theorems 4.1–5.1 are readily exploited to conclude an asymptotic risk efficiency property and second-order regret expansion, respectively. A number of other interesting illustrations are highlighted where it is possible to verify the conclusions from Theorems 4.1–5.1 more directly with less stringent assumptions on the pilot sample size.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":"38 1","pages":"480 - 502"},"PeriodicalIF":0.8,"publicationDate":"2019-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2019.1686885","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45421026","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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