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Why is it important?
The problem of solving the ordered one-way analysis of variance (ANOVA) (which consists of comparing a set of normal means) with restricted Type I and Type II error rates is considered in this paper. This case is more complicated than unordered one-way ANOVA because the detection of the monotonicity of means restrictions is necessary. To solve this issue, one of the possible formulations of the constrained Bayesian method (CBM) is applied here using the concept of directional hypotheses. The cases of known and unknown variances are examined. For unknown variances, the maximum likelihood ratio and Stein’s Methods were used to overcome the problem connected with the complexity of hypotheses. The correctness of the developed methods and high quality (in comparison with existing methods) of obtained results were demonstrated by computing results of the simulated concrete scenarios. Moreover, the offered method controlled not only one Type of error, as methods do, but both Type I and Type II methods.
Perspectives
The problem of testing hypotheses concerning the means of a set of normally distributed random variables at the restriction of both Type I and Type II error rates are considered in this paper. To solve the stated problem, one of the possible statements of CBM is used. In order to overcome the difficulties related to the complexity of testing hypotheses caused by not knowing the values of the variances of random variables, the maximum likelihood and Stein’s methods are used. The ability to limit Type I and Type II error rates and other relevant criteria to the required level is shown for the developed algorithms. The latter is confirmed by the calculation results of practical examples by simulating different scenarios.
Professor Kartlos Joseph Kachiashvili
Georgian Tachnical University
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This page is a summary of: Solving ANOVA problem with restricted Type Ⅰ and Type Ⅱ error rates, AIMS Mathematics, January 2025, Tsinghua University Press,
DOI: 10.3934/math.2025109.
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