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":null,"pages":null},"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":null,"pages":null},"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}
Pub Date : 2019-08-03DOI: 10.1080/07474946.2021.1847924
A. Muravlev, M. Urusov, M. Zhitlukhin
Abstract We consider an optimal control problem where a Brownian motion with drift is sequentially observed and the sign of the drift coefficient changes at jump times of a symmetric two-state Markov process. The Markov process itself is not observable, and the problem consists of finding a {−1, 1}-valued process that tracks the unobservable process as closely as possible. We present an explicit construction of such a process.
{"title":"Sequential tracking of an unobservable two-state Markov process under Brownian noise","authors":"A. Muravlev, M. Urusov, M. Zhitlukhin","doi":"10.1080/07474946.2021.1847924","DOIUrl":"https://doi.org/10.1080/07474946.2021.1847924","url":null,"abstract":"Abstract We consider an optimal control problem where a Brownian motion with drift is sequentially observed and the sign of the drift coefficient changes at jump times of a symmetric two-state Markov process. The Markov process itself is not observable, and the problem consists of finding a {−1, 1}-valued process that tracks the unobservable process as closely as possible. We present an explicit construction of such a process.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2021.1847924","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41414373","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 : 2019-07-05DOI: 10.1080/07474946.2020.1766926
Kexuan Li, Aleksey S. Polunchenko
Abstract For the classical Shiryaev-Roberts martingale diffusion considered on the interval where A > 0 is a given absorbing boundary, it is shown that the rate of convergence of the diffusion’s quasi-stationary cumulative distribution function (c.d.f.), to its stationary c.d.f., H(x), as is no worse than uniformly in The result is established explicitly by constructing new tight lower- and upper-bounds for using certain latest monotonicity properties of the modified Bessel K function involved in the exact closed-form formula for recently obtained by Polunchenko (2017b).
{"title":"On the convergence rate of the quasi- to stationary distribution for the Shiryaev-Roberts diffusion","authors":"Kexuan Li, Aleksey S. Polunchenko","doi":"10.1080/07474946.2020.1766926","DOIUrl":"https://doi.org/10.1080/07474946.2020.1766926","url":null,"abstract":"Abstract For the classical Shiryaev-Roberts martingale diffusion considered on the interval where A > 0 is a given absorbing boundary, it is shown that the rate of convergence of the diffusion’s quasi-stationary cumulative distribution function (c.d.f.), to its stationary c.d.f., H(x), as is no worse than uniformly in The result is established explicitly by constructing new tight lower- and upper-bounds for using certain latest monotonicity properties of the modified Bessel K function involved in the exact closed-form formula for recently obtained by Polunchenko (2017b).","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2020.1766926","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49452546","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 : 2019-07-03DOI: 10.1080/07474946.2019.1648919
Z. Djeridi, H. Merabet
Abstract In this article, we propose a hybrid-Bayesian frequentist approach using a Bayesian sequential prediction of the index of satisfaction. For interim analysis that addresses prediction hypothesis, such as futility monitoring with delayed outcomes, the prediction of satisfaction properly accounts for the amount of data remaining to be observed in a clinical trial and has the flexibility to incorporate additional information via auxiliary variables. The prediction of satisfaction design guarantees the type I error rate and does not require intensive computation or comprehensive simulation. The design is retrospectively applied to a lung cancer clinical trial.
{"title":"A hybrid Bayesian-frequentist predictive design for monitoring multi-stage clinical trials","authors":"Z. Djeridi, H. Merabet","doi":"10.1080/07474946.2019.1648919","DOIUrl":"https://doi.org/10.1080/07474946.2019.1648919","url":null,"abstract":"Abstract In this article, we propose a hybrid-Bayesian frequentist approach using a Bayesian sequential prediction of the index of satisfaction. For interim analysis that addresses prediction hypothesis, such as futility monitoring with delayed outcomes, the prediction of satisfaction properly accounts for the amount of data remaining to be observed in a clinical trial and has the flexibility to incorporate additional information via auxiliary variables. The prediction of satisfaction design guarantees the type I error rate and does not require intensive computation or comprehensive simulation. The design is retrospectively applied to a lung cancer clinical trial.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2019.1648919","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45734394","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 : 2019-07-03DOI: 10.1080/07474946.2019.1648933
T. N. Sriram, S. Samadi
Abstract This article revisits the problem of sequential point estimation of the autogressive parameter in an autoregressive model of order 1, where the errors are independent and identically distributed with mean 0 and unknown variance . This problem was originally considered in Sriram (1988), where first-order efficiency properties and a second-order expansion for the expected value of a stopping rule were established. Here, we obtain an asymptotic expression for the so-called regret due to not knowing σ, as the cost of estimation error tends to infinity. Under suitable assumptions, our extensive analysis shows that all but one term in the regret are asymptotically bounded. If the errors have a bounded support, however, then the regret remains asymptotically bounded. Finally, we illustrate the performance of our sequential procedure and the associated regret for well-known blowfly data (Nicholson, 1950) and Internet traffic data using the residual bootstrap method for autoregressive models.
{"title":"Second-order analysis of regret for sequential estimation of the autoregressive parameter in a first-order autoregressive model","authors":"T. N. Sriram, S. Samadi","doi":"10.1080/07474946.2019.1648933","DOIUrl":"https://doi.org/10.1080/07474946.2019.1648933","url":null,"abstract":"Abstract This article revisits the problem of sequential point estimation of the autogressive parameter in an autoregressive model of order 1, where the errors are independent and identically distributed with mean 0 and unknown variance . This problem was originally considered in Sriram (1988), where first-order efficiency properties and a second-order expansion for the expected value of a stopping rule were established. Here, we obtain an asymptotic expression for the so-called regret due to not knowing σ, as the cost of estimation error tends to infinity. Under suitable assumptions, our extensive analysis shows that all but one term in the regret are asymptotically bounded. If the errors have a bounded support, however, then the regret remains asymptotically bounded. Finally, we illustrate the performance of our sequential procedure and the associated regret for well-known blowfly data (Nicholson, 1950) and Internet traffic data using the residual bootstrap method for autoregressive models.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2019.1648933","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49453386","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 : 2019-07-03DOI: 10.1080/07474946.2019.1648923
Marlo Brown
Abstract We look at a Poisson process where the arrival rates change from a known λ1 to a known λ2. Whereas in most of the literature the change-point is abrupt, we model the more realistic assumption that states that the change happens gradually over a period of time η where η is known. We calculate the probability that the change has started and completed. We also look at optimal stopping rules assuming that there is a cost for a false alarm and a cost per time unit to stop early. We conclude with some numerical results.
{"title":"Monitoring a Poisson process subject to gradual changes in the arrival rates","authors":"Marlo Brown","doi":"10.1080/07474946.2019.1648923","DOIUrl":"https://doi.org/10.1080/07474946.2019.1648923","url":null,"abstract":"Abstract We look at a Poisson process where the arrival rates change from a known λ1 to a known λ2. Whereas in most of the literature the change-point is abrupt, we model the more realistic assumption that states that the change happens gradually over a period of time η where η is known. We calculate the probability that the change has started and completed. We also look at optimal stopping rules assuming that there is a cost for a false alarm and a cost per time unit to stop early. We conclude with some numerical results.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2019.1648923","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45948903","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 : 2019-07-03DOI: 10.1080/07474946.2019.1648920
K. Zamba, A. Adekpedjou
Abstract This article provides a goodness-of-fit test for the distribution function or the survival function in a recurrent event setting, when the inter-event time parametric structure is estimated from the observed data. Of concern is the null hypothesis that the inter-event time distribution is absolutely continuous and belongs to a parametric family , where the q-dimensional parameter space is neither known nor specified. We proposed a Khmaladze martingale-transformed type of test (Khmaladze, 1981), adapted to recurrent events. The test statistic combines two likelihood sources of estimation to form a parametric empirical process: (1) a product-limit nonparametric maximum likelihood estimator (NPMLE; Peña et al., 2001a) that is a consistent estimator of F, say, and (2) a point process likelihood estimator (Jacod, 1974/1975). These estimators are combined to construct a Kolmogorov-Smirnov (KS) type of test (Kolmogorov 1933; Smirnov, 1933). Empirical process and martingale weak convergence frameworks are utilized for theoretical derivations and motivational justification of the proposed transformation. A simulation study is conducted for performance assessment, and the test is applied to a problem investigated by Proschan (1963) on air-conditioning failure in a fleet of Boeing 720 jets.
摘要本文对递归事件环境中的分布函数或生存函数进行了拟合优度检验，当根据观测数据估计事件间时间参数结构时。值得关注的是零假设，即事件间时间分布是绝对连续的，并且属于参数族，其中q维参数空间既不已知也不指定。我们提出了一种适应于复发事件的Khmaladze鞅转换型测试（Khmaladz，1981）。检验统计量结合了两个估计的似然源，形成了一个参数经验过程：（1）乘积极限非参数最大似然估计量（NPMLE；Peña et al.，2001a），例如，它是F的一致估计量；（2）点过程似然估计器（Jacobd，1974/1975）。将这些估计量组合起来构建Kolmogorov-Smirnov（KS）类型的检验（Kolmogorov 1933；Smirnov，1933）。利用经验过程和鞅弱收敛框架对所提出的变换进行理论推导和动机论证。对性能评估进行了模拟研究，并将该测试应用于Proschan（1963）研究的波音720喷气机队空调故障问题。
{"title":"A Khmaladze-transformed test of fit with ML estimation in the presence of recurrent events","authors":"K. Zamba, A. Adekpedjou","doi":"10.1080/07474946.2019.1648920","DOIUrl":"https://doi.org/10.1080/07474946.2019.1648920","url":null,"abstract":"Abstract This article provides a goodness-of-fit test for the distribution function or the survival function in a recurrent event setting, when the inter-event time parametric structure is estimated from the observed data. Of concern is the null hypothesis that the inter-event time distribution is absolutely continuous and belongs to a parametric family , where the q-dimensional parameter space is neither known nor specified. We proposed a Khmaladze martingale-transformed type of test (Khmaladze, 1981), adapted to recurrent events. The test statistic combines two likelihood sources of estimation to form a parametric empirical process: (1) a product-limit nonparametric maximum likelihood estimator (NPMLE; Peña et al., 2001a) that is a consistent estimator of F, say, and (2) a point process likelihood estimator (Jacod, 1974/1975). These estimators are combined to construct a Kolmogorov-Smirnov (KS) type of test (Kolmogorov 1933; Smirnov, 1933). Empirical process and martingale weak convergence frameworks are utilized for theoretical derivations and motivational justification of the proposed transformation. A simulation study is conducted for performance assessment, and the test is applied to a problem investigated by Proschan (1963) on air-conditioning failure in a fleet of Boeing 720 jets.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2019.1648920","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47419143","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 : 2019-07-03DOI: 10.1080/07474946.2019.1649347
E. Mahmoudi, Ashkan Khalifeh, V. Nekoukhou
Abstract In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.
{"title":"Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution","authors":"E. Mahmoudi, Ashkan Khalifeh, V. Nekoukhou","doi":"10.1080/07474946.2019.1649347","DOIUrl":"https://doi.org/10.1080/07474946.2019.1649347","url":null,"abstract":"Abstract In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2019.1649347","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44062906","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 : 2019-07-03DOI: 10.1080/07474946.2019.1648926
Ajit Chaturvedi, Sudeep R. Bapat, Neeraj Joshi
Abstract In this article, we have estimated the mean vector of a multivariate normal population by using a k-stage sequential estimation procedure. Point estimation as well as confidence region estimation is done. Second-order approximations are obtained in both the cases. In case of minimum risk point estimation of , negative regret is achieved.
{"title":"A k-stage procedure for estimating the mean vector of a multivariate normal population","authors":"Ajit Chaturvedi, Sudeep R. Bapat, Neeraj Joshi","doi":"10.1080/07474946.2019.1648926","DOIUrl":"https://doi.org/10.1080/07474946.2019.1648926","url":null,"abstract":"Abstract In this article, we have estimated the mean vector of a multivariate normal population by using a k-stage sequential estimation procedure. Point estimation as well as confidence region estimation is done. Second-order approximations are obtained in both the cases. In case of minimum risk point estimation of , negative regret is achieved.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07474946.2019.1648926","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45194257","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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