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In this paper, we deal with sequential testing of multiple hypotheses. In the general scheme of construction of optimal tests based on the backward induction, we propose a modification which provides a simplified (generally speaking, suboptimal) version of the optimal test, for any particular criterion of optimization. We call this DBC version (the one with Dropped Backward Control) of the optimal test. In particular, for the case of two simple hypotheses, dropping backward control in the Bayesian test produces the classical sequential probability ratio test (SPRT). Similarly, dropping backward control in the modified Kiefer-Weiss solutions produces Lorden's 2-SPRTs . In the case of more than two hypotheses, we obtain in this way new classes of sequential multi-hypothesis tests, and investigate their properties. The efficiency of the DBC-tests is evaluated with respect to the optimal Bayesian multi-hypothesis test and with respect to the matrix sequential probability ratio test (MSPRT) by Armitage. In a multihypothesis variant of the Kiefer-Weiss problem for binomial proportions the performance of the DBC-test is numerically compared with that of the exact solution. In a model of normal observations with a linear trend, the performance of of the DBC-test is numerically compared with that of the MSPRT. Some other numerical examples are presented. In all the cases the proposed tests exhibit a very high efficiency with respect to the optimal tests (more than 99.3\% when sampling from Bernoulli populations) and/or with respect to the MSPRT (even outperforming the latter in some scenarios)., Comment: 16 pages, 4 tables

In this paper 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 ESS's 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 which is 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, 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 minimize the Lagrangian function and for the numerical evaluation of test characteristics like the error probabilities and the ESS, and other related. 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 is proposed, and is applied for a particular case of three hypotheses in the Bernoulli model., Comment: 21 pages, 2 tables

In this paper, we deal with problems of testing hypotheses in the framework of sequential statistical analysis. The main concern is the optimal design and performance evaluation of sampling plans in the Kiefer-Weiss problems. For the observations which follow a discrete exponential family, we provide algorithms for optimal design in the modified Kiefer-Weiss problem, and obtain formulas for evaluating their performance, calculating operating characteristic function, average sample number, and some related characteristics. These formulas cover, as a particular case, the sequential probability ratio tests (SPRT) and their truncated versions, as well as optimal finite-horizon sequential tests. On the basis of the developed algorithms we propose a method of construction of optimal tests and their performance evaluation for the original Kiefer-Weiss problem. All the algorithms are implemented as functions in the R programming language and can be downloaded from {\tt https://github.com/tosinabase/Kiefer-Weiss}, where the functions for the binomial, Poisson, and negative binomial distributions are readily available. Finally, we make numerical comparisons of the Kiefer-Weiss solution with the SPRT and the fixed-sample-size test having the same levels of the error probabilities., Comment: 20 pages, 4 tables, accompanying program code provided

We present a computational approach to 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., Comment: 23 pages, 5 Tables, 3 Figures