Solving ANOVA problem with restricted Type Ⅰ and Type Ⅱ error rates

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.