In this paper we deal with the problem of sequential testing of multiple hypotheses. The main goal is minimising the expected sample size (ESS) under restrictions on the error probabilities. We use a variant of the method of Lagrange multipliers which is based on the minimisation of an auxiliary objective function (called Lagrangian). This function is defined as a weighted sum of all the test characteristics we are interested in: the error probabilities and the ESSs evaluated at some points of interest. In this paper, we use a definition of the Lagrangian function involving the ESS evaluated at any finite number of fixed parameter points (not necessarily those representing the hypotheses). Then we develop a computer-oriented method of minimisation of the Lagrangian function, that provides, depending on the specific choice of the parameter points, optimal tests in different concrete settings, like 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 minimise the Lagrangian function and for the numerical evaluation of test characteristics like the error probabilities and the ESS, and other related. 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.

翻译：本文研究多假设序贯检验问题。主要目标是在误差概率受限条件下最小化期望样本量。我们采用拉格朗日乘子法的一种变体，该方法基于辅助目标函数（称为拉格朗日函数）的最小化。该函数定义为所有感兴趣检验特征量的加权和：误差概率以及在某些关注点上评估的期望样本量。本文采用的定义涉及在任意有限个固定参数点（不一定是代表假设的参数点）上评估的期望样本量。随后我们开发了一种计算机导向的拉格朗日函数最小化方法，根据参数点的具体选择，可在贝叶斯、基弗-韦斯等不同具体设定中提供最优检验。为示例所提方法在伯努利总体抽样这一特例中的应用，我们开发了一组计算机算法，用于设计使拉格朗日函数最小化的序贯检验，并数值评估检验特征量（如误差概率、期望样本量及其他相关量）。针对伯努利模型，我们围绕三个假设的特例进行了一系列关于序贯多假设检验最优性的计算机评估，并与矩阵序贯概率比检验进行了数值比较。