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AIMS Mathematics

2025, Volume 10, Issue 2: 2347-2374. doi: 10.3934/math.2025109
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Solving ANOVA problem with restricted Type Ⅰ and Type Ⅱ error rates

  • 1.
    Faculty of Informatics and Control Systems, Georgian Technical University, Tbilisi, 0160, Georgia
  • 2.
    Department of Probability Theory and Mathematical Statistics of the Ⅰ. Vekua Institute of Applied Mathematics, Tbilisi State University, Tbilisi, Georgia
  • 3.
    Department of Informatics of the Muskhelishvili Institute of Computational Mathematics, Georgian Technical University, Tbilisi, Georgia
  • 4.
    Department of Mathematics, Indian Institute of Technology, Kharagpur, W.B., India
  • 5.
    Department of Population Health Sciences, Medical College of Georgia, Augusta University, Augusta, Georgia, USA
  • 6.
    Department of Statistics, Middle East Technical University, Ankara, Turkey
  • Received: 16 November 2024 Revised: 25 December 2024 Accepted: 08 January 2025 Published: 10 February 2025

  • MSC : 62F15, 62F03

  • 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.

    Citation: Kartlos J. Kachiashvili, Joseph K. Kachiashvili, Ashis SenGupta. Solving ANOVA problem with restricted Type Ⅰ and Type Ⅱ error rates[J]. AIMS Mathematics, 2025, 10(2): 2347-2374. doi: 10.3934/math.2025109

    Related Papers:

  • 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.



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