New methods for statistical decision making in conditions of a limited volume of observations and with a prioriy parametric uncertainty

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A new generalized adaptive algorithm for learning to make statistical decisions for exponential families of distributions with a priori parametric uncertainty in conditions of small samples has been developed. A generalized decision rule is presented, obtained by estimating unknown parameters of distributions, as well as a decision rule that satisfies the necessary optimality conditions: constancy of the average probability of a type I error and unbiasedness. Specific decision procedures for partial distributions obtained from a generalized algorithm are considered. Numerical examples are given. The effectiveness of the developed optimal procedure for small samples is shown.

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作者简介

F. Mkrtchyan

Fryazino Branch Kotelnikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences

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Email: ferd47@mail.ru
俄罗斯联邦, Fryazino, Moscow region, 141190 Russia

参考

  1. Арманд Н.А., Крапивин В.Ф., Мкртчян Ф.А. Методы обработки данных радиофизического исследования окружающей среды. М.: Наука, 1987.
  2. Мкртчян Ф.А. Оптимальное различение сигналов и проблемы мониторинга. М.: Наука, 1982.
  3. Mkrtchyan F.A., Varotsos C.A. // Water, Air, & Soil Pollution. 2018. V. 229. № 8. Article No. 273.
  4. Данков П.П. // РЭ. 1965. Т. 10. № 10. С. 1774.
  5. Леман Э. Проверка статистических гипотез. М.: Наука, 1979.

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2. Fig. 1. Acceptance regions of hypotheses for the classical decision rule.

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3. Fig. 2. Hypothesis acceptance regions for the optimal decision rule; the hypothesis acceptance region is shaded.

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4. Fig. 3. Graphs of the probabilities of errors of the first and second kind for the classical decision procedure.

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5. Fig. 4. Graphs of the probabilities of errors of the first and second kind for the optimal decision procedure.

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6. Fig. 5. The ratio of decision procedures for α₀ = 0.05 (1), 0.1 (2) and 0.2 (3).

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