A numerical approach to sequential multi-hypothesis testing for Bernoulli model

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY
A. Novikov
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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.
伯努利模型序贯多假设检验的数值方法
摘要本文研究了多假设的序贯检验问题。主要目标是在误差概率的限制下最小化期望样本量。在误差概率的限制下,我们将在参数空间中某些兴趣点处评估的ESSs的加权和作为最小化准则,目的是使ESSs最小化。我们使用拉格朗日乘子方法的一种变体,该方法基于辅助目标函数(称为拉格朗日)的最小化,将目标函数与限制结合起来,并采用一些称为乘子的常数。随后,使用乘法器使解符合约束条件。我们开发了一种面向计算机的拉格朗日函数最小化方法,该方法根据参数点的具体选择,在不同的具体设置(如贝叶斯、Kiefer-Weiss和其他设置)中提供最佳测试。为了举例说明从伯努利总体中抽样的特定情况下提出的方法,我们开发了一套计算机算法,用于设计使拉格朗日函数最小化的顺序测试,并用于对测试特征(如误差概率和ESS)进行数值评估。我们用R编程语言实现算法。该程序代码可在GitHub公共存储库中获得。对于伯努利模型,我们对顺序多假设检验的最优性进行了一系列计算机评估,在三个假设的特定情况下。并与矩阵序列概率比检验进行了数值比较。提出了一种求解多假设Kiefer-Weiss问题的方法,并应用于伯努利模型中三个假设的特殊情况。
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来源期刊
CiteScore
1.40
自引率
12.50%
发文量
20
期刊介绍: The purpose of Sequential Analysis is to contribute to theoretical and applied aspects of sequential methodologies in all areas of statistical science. Published papers highlight the development of new and important sequential approaches. Interdisciplinary articles that emphasize the methodology of practical value to applied researchers and statistical consultants are highly encouraged. Papers that cover contemporary areas of applications including animal abundance, bioequivalence, communication science, computer simulations, data mining, directional data, disease mapping, environmental sampling, genome, imaging, microarrays, networking, parallel processing, pest management, sonar detection, spatial statistics, tracking, and engineering are deemed especially important. Of particular value are expository review articles that critically synthesize broad-based statistical issues. Papers on case-studies are also considered. All papers are refereed.
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