Pub Date : 2021-10-10DOI: 10.1080/07474946.2022.2070212
A. Novikov, Andrei Novikov, Fahil Farkhshatov
Abstract We present a computational approach to the solution of the Kiefer-Weiss problem. Algorithms for construction of the optimal sampling plans and evaluation of their performance are proposed. In the particular case of Bernoulli observations, the proposed algorithms are implemented in the form of R program code. Using the developed computer program, we numerically compare the optimal tests with the respective sequential probability ratio test (SPRT) and the fixed sample size test for a wide range of hypothesized values and type I and type II errors. The results are compared with those of D. Freeman and L. Weiss (Journal of the American Statistical Association, 59, 1964). The R source code for the algorithms of construction of optimal sampling plans and evaluation of their characteristics is available at https://github.com/tosinabase/Kiefer-Weiss.
{"title":"A computational approach to the Kiefer-Weiss problem for sampling from a Bernoulli population","authors":"A. Novikov, Andrei Novikov, Fahil Farkhshatov","doi":"10.1080/07474946.2022.2070212","DOIUrl":"https://doi.org/10.1080/07474946.2022.2070212","url":null,"abstract":"Abstract We present a computational approach to the solution of the Kiefer-Weiss problem. Algorithms for construction of the optimal sampling plans and evaluation of their performance are proposed. In the particular case of Bernoulli observations, the proposed algorithms are implemented in the form of R program code. Using the developed computer program, we numerically compare the optimal tests with the respective sequential probability ratio test (SPRT) and the fixed sample size test for a wide range of hypothesized values and type I and type II errors. The results are compared with those of D. Freeman and L. Weiss (Journal of the American Statistical Association, 59, 1964). The R source code for the algorithms of construction of optimal sampling plans and evaluation of their characteristics is available at https://github.com/tosinabase/Kiefer-Weiss.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42289505","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 : 2021-10-08DOI: 10.1080/07474946.2022.2070213
A. Novikov, X. I. Popoca-Jiménez
Abstract We consider sequential hypothesis testing based on observations that are received in groups of random size. The observations are assumed to be independent both within and between the groups. We assume that the group sizes are independent and their distributions are known and that the groups are formed independent of the observations. We are concerned with a problem of testing a simple hypothesis against a simple alternative. For any (group) sequential test, we take into account the following three characteristics: its type I and type II error probabilities and the average cost of observations. Under mild conditions, we characterize the structure of sequential tests minimizing the average cost of observations among all sequential tests whose type I and type II error probabilities do not exceed some prescribed levels.
{"title":"Optimal group sequential tests with groups of random size","authors":"A. Novikov, X. I. Popoca-Jiménez","doi":"10.1080/07474946.2022.2070213","DOIUrl":"https://doi.org/10.1080/07474946.2022.2070213","url":null,"abstract":"Abstract We consider sequential hypothesis testing based on observations that are received in groups of random size. The observations are assumed to be independent both within and between the groups. We assume that the group sizes are independent and their distributions are known and that the groups are formed independent of the observations. We are concerned with a problem of testing a simple hypothesis against a simple alternative. For any (group) sequential test, we take into account the following three characteristics: its type I and type II error probabilities and the average cost of observations. Under mild conditions, we characterize the structure of sequential tests minimizing the average cost of observations among all sequential tests whose type I and type II error probabilities do not exceed some prescribed levels.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45264470","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 : 2021-10-02DOI: 10.1080/07474946.2021.2010409
N. Mulgan
Abstract Sequential analysis for the purposes of possibly stopping a trial early is important whenever a result must be obtained as quickly as possible for public health, economic, or other reasons. A dominant research stream since the middle of the 20th century has been the challenge of generalizing Wald’s sequential probability ratio test (SPRT) to include composite alternative hypotheses. This article offers an alternative for a binomial distribution by constructing a single-parameter family of triangular rejection regions for the null hypothesis using a generation function argument in the two-dimensional space of successes versus failures. The result is algebraically equivalent to one line of the SPRT with unit power and with the second value of the binomial parameter as the undetermined parameter. Rounding then discretizes the line to the grid of two-dimensional integers and classical results for arbitrary stopping conditions are used to give expressions for the estimator, of the binomial parameter and its variance The choice can be made by the practitioner in terms of their appetite for risk or more formally via a power analysis. An example is given of an ecological study where a quick binary decision was required and this desire had to be weighed against robustness of the results.
{"title":"Binomial early stopping times","authors":"N. Mulgan","doi":"10.1080/07474946.2021.2010409","DOIUrl":"https://doi.org/10.1080/07474946.2021.2010409","url":null,"abstract":"Abstract Sequential analysis for the purposes of possibly stopping a trial early is important whenever a result must be obtained as quickly as possible for public health, economic, or other reasons. A dominant research stream since the middle of the 20th century has been the challenge of generalizing Wald’s sequential probability ratio test (SPRT) to include composite alternative hypotheses. This article offers an alternative for a binomial distribution by constructing a single-parameter family of triangular rejection regions for the null hypothesis using a generation function argument in the two-dimensional space of successes versus failures. The result is algebraically equivalent to one line of the SPRT with unit power and with the second value of the binomial parameter as the undetermined parameter. Rounding then discretizes the line to the grid of two-dimensional integers and classical results for arbitrary stopping conditions are used to give expressions for the estimator, of the binomial parameter and its variance The choice can be made by the practitioner in terms of their appetite for risk or more formally via a power analysis. An example is given of an ecological study where a quick binary decision was required and this desire had to be weighed against robustness of the results.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41853745","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 : 2021-10-02DOI: 10.1080/07474946.2021.2010417
B. Levin, C. Leu
Abstract We introduce and discuss some key inequalities that underlie the lower bound formula for the probability of lattice events in the Levin-Robbins-Leu family of sequential subset selection procedures for binary outcomes. The present work combines the notion of lattice events—as previously discussed for the nonadaptive member of the family—with the positive cumulative sum property for the adaptive members—as previously discussed for the special lattice event of correct selection, thereby extending the key inequality to its broadest scope.
{"title":"A key inequality for lower bound formulas for lattice event probabilities","authors":"B. Levin, C. Leu","doi":"10.1080/07474946.2021.2010417","DOIUrl":"https://doi.org/10.1080/07474946.2021.2010417","url":null,"abstract":"Abstract We introduce and discuss some key inequalities that underlie the lower bound formula for the probability of lattice events in the Levin-Robbins-Leu family of sequential subset selection procedures for binary outcomes. The present work combines the notion of lattice events—as previously discussed for the nonadaptive member of the family—with the positive cumulative sum property for the adaptive members—as previously discussed for the special lattice event of correct selection, thereby extending the key inequality to its broadest scope.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48287873","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 : 2021-10-02DOI: 10.1080/07474946.2021.2010403
M. Haneche, K. Djaballah, K. Khaldi
Abstract The aim of this work is to approximate the trajectory solution of parabolic partial differential equations (PDEs) by the probabilistic method. This method is based on the representation of Feynman-Kac and Monte Carlo methods. As an alternative to classical Monte Carlo, here we employ quasi–Monte Carlo methods and propose some solutions to the problem of using this alternative through a more efficient algorithm than the classics.
{"title":"An algorithm for probabilistic solution of parabolic PDEs","authors":"M. Haneche, K. Djaballah, K. Khaldi","doi":"10.1080/07474946.2021.2010403","DOIUrl":"https://doi.org/10.1080/07474946.2021.2010403","url":null,"abstract":"Abstract The aim of this work is to approximate the trajectory solution of parabolic partial differential equations (PDEs) by the probabilistic method. This method is based on the representation of Feynman-Kac and Monte Carlo methods. As an alternative to classical Monte Carlo, here we employ quasi–Monte Carlo methods and propose some solutions to the problem of using this alternative through a more efficient algorithm than the classics.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42936358","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 : 2021-10-02DOI: 10.1080/07474946.2021.2010407
Neeraj Joshi, Sudeep R. Bapat
Abstract In this article, we develop a general class of purely sequential procedures and obtain the associated first- and second-order asymptotics for the expected sample size and regret. We establish that many estimation and ranking and selection problems can be handled with the help of the proposed class of sequential procedures. A brief simulation analysis is carried out in support of the accuracy of our proposed sequential methodology and a real data set from environment study is included to demonstrate the practical utility.
{"title":"On a class of purely sequential procedures with applications to estimation and ranking and selection problems","authors":"Neeraj Joshi, Sudeep R. Bapat","doi":"10.1080/07474946.2021.2010407","DOIUrl":"https://doi.org/10.1080/07474946.2021.2010407","url":null,"abstract":"Abstract In this article, we develop a general class of purely sequential procedures and obtain the associated first- and second-order asymptotics for the expected sample size and regret. We establish that many estimation and ranking and selection problems can be handled with the help of the proposed class of sequential procedures. A brief simulation analysis is carried out in support of the accuracy of our proposed sequential methodology and a real data set from environment study is included to demonstrate the practical utility.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48090325","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 : 2021-10-02DOI: 10.1080/07474946.2021.2010411
Ying-Ying Zhang, Tengzhong Rong, Man-Man Li
Abstract Assuming normality for the prior and the likelihood, we calculate the rejection region, the power or the conditional power, and the predictive power or the conditional predictive power of one-sided hypotheses with a nonzero threshold that corresponds to a noninferiority test for two-arm trials for five different scenarios, which are nonsequential trials with classical power and Bayesian power and sequential trials with hybrid predictions, Bayesian predictions, and classical predictions. The rejection regions and the powers of one-sided hypotheses with a zero threshold that corresponds to a superiority test for two-arm trials are also obtained. Then the calculations of the various powers are illustrated through two examples. The article can be regarded as a reference manual for researchers interested in power calculations of one-sided hypotheses with a nonzero or zero threshold for the five different scenarios assuming normality for the prior and the likelihood.
{"title":"Analytical calculations of various powers assuming normality","authors":"Ying-Ying Zhang, Tengzhong Rong, Man-Man Li","doi":"10.1080/07474946.2021.2010411","DOIUrl":"https://doi.org/10.1080/07474946.2021.2010411","url":null,"abstract":"Abstract Assuming normality for the prior and the likelihood, we calculate the rejection region, the power or the conditional power, and the predictive power or the conditional predictive power of one-sided hypotheses with a nonzero threshold that corresponds to a noninferiority test for two-arm trials for five different scenarios, which are nonsequential trials with classical power and Bayesian power and sequential trials with hybrid predictions, Bayesian predictions, and classical predictions. The rejection regions and the powers of one-sided hypotheses with a zero threshold that corresponds to a superiority test for two-arm trials are also obtained. Then the calculations of the various powers are illustrated through two examples. The article can be regarded as a reference manual for researchers interested in power calculations of one-sided hypotheses with a nonzero or zero threshold for the five different scenarios assuming normality for the prior and the likelihood.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46264216","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 : 2021-10-02DOI: 10.1080/07474946.2021.2010414
Pritam Sarkar, U. Bandyopadhyay
Abstract In this article we concentrate on fixed accuracy intervals of the common variance when the data arise from a Gaussian process with order 1 autoregressive covariance structure. Our approach includes the maximum likelihood method and least squares method for estimating the parameters in this process. We provide necessary asymptotic results and carry out numerical evaluations.
{"title":"On sequential confidence interval in a stationary Gaussian process","authors":"Pritam Sarkar, U. Bandyopadhyay","doi":"10.1080/07474946.2021.2010414","DOIUrl":"https://doi.org/10.1080/07474946.2021.2010414","url":null,"abstract":"Abstract In this article we concentrate on fixed accuracy intervals of the common variance when the data arise from a Gaussian process with order 1 autoregressive covariance structure. Our approach includes the maximum likelihood method and least squares method for estimating the parameters in this process. We provide necessary asymptotic results and carry out numerical evaluations.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48963161","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 : 2021-07-03DOI: 10.1080/07474946.2021.1940493
Radislav Vaisman
Abstract The efficient evaluation of high-dimensional integrals is important from both theoretical and practical points of view. In particular, multidimensional integration plays a central role in Bayesian inference, statistical physics, data science, and machine learning. However, due to the curse of dimensionality, deterministic numerical methods are inefficient in the high-dimensional setting. Consequentially, for many practical problems one must resort to approximate estimation techniques such as Monte Carlo methods. In this article, we introduce a novel sequential Monte Carlo algorithm called stratified splitting. The method provides unbiased estimates and can handle various integrand types including indicator functions, which are important for rare-event probability estimation problems. We provide rigorous analysis of the efficiency of the proposed method and present a numerical demonstration of the algorithmic performance when applied to practical application domains. Our numerical experiments suggest that the stratified splitting method is capable of delivering accurate results for a variety of integration problems while requiring reasonable computational effort.
{"title":"Sequential stratified splitting for efficient Monte Carlo integration","authors":"Radislav Vaisman","doi":"10.1080/07474946.2021.1940493","DOIUrl":"https://doi.org/10.1080/07474946.2021.1940493","url":null,"abstract":"Abstract The efficient evaluation of high-dimensional integrals is important from both theoretical and practical points of view. In particular, multidimensional integration plays a central role in Bayesian inference, statistical physics, data science, and machine learning. However, due to the curse of dimensionality, deterministic numerical methods are inefficient in the high-dimensional setting. Consequentially, for many practical problems one must resort to approximate estimation techniques such as Monte Carlo methods. In this article, we introduce a novel sequential Monte Carlo algorithm called stratified splitting. The method provides unbiased estimates and can handle various integrand types including indicator functions, which are important for rare-event probability estimation problems. We provide rigorous analysis of the efficiency of the proposed method and present a numerical demonstration of the algorithmic performance when applied to practical application domains. Our numerical experiments suggest that the stratified splitting method is capable of delivering accurate results for a variety of integration problems while requiring reasonable computational effort.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45630883","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 : 2021-07-03DOI: 10.1080/07474946.2021.1940504
Marlo Brown
Abstract We look at a Poisson process where the arrival rates change from λ 1 to λ 2. We will assume that the arrival rates both before and after the change are unknown. We also assume that this change does not happen abruptly but 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.
{"title":"Monitoring a Poisson process subject to gradual changes in the arrival rates where the arrival rates are unknown","authors":"Marlo Brown","doi":"10.1080/07474946.2021.1940504","DOIUrl":"https://doi.org/10.1080/07474946.2021.1940504","url":null,"abstract":"Abstract We look at a Poisson process where the arrival rates change from λ 1 to λ 2. We will assume that the arrival rates both before and after the change are unknown. We also assume that this change does not happen abruptly but 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.","PeriodicalId":48879,"journal":{"name":"Sequential Analysis-Design Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49371825","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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