Study on approximation for doubly non-central F distribution based on first two moments

Shingo Inoue, Ikuo Arizono, Ryosuke Tomohiro, Yasuhiko Takemoto, Akihiro Kanagawa

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A SN ratio is famous as a criterion for parameter design using the Taguchi method. In recent years, it has been revealed that the statistical feature of the SN ratio is described by a doubly non-central F distribution. Accordingly, the characteristics of the doubly non-central F distribution are gaining attention again. However, it is extremely difficult to theoretically and exactly evaluate a percentile for the doubly non-central F distribution since its probability density function is very complicated. Therefore, approximation techniques for evaluating the percentile have been conventionally studied. One of them, a method for approximating the doubly non-central F distribution using central F distribution based on the first three approximate moments in the doubly non-central F distribution has been devised. Further, an evaluation of the doubly non-central F distribution percentile based on Cornish-Fisher expansion using those approximate values of first three moments has been proposed. In this study, we calculate the accurate values of moments in doubly non-central F distribution, and then propose a new method for evaluating the percentile of the distribution using the accurate values of the first two moments. Through some numerical simulations, the effectiveness of our new proposal in this study is confirmed. In addition, as an application of our proposal, the confidence interval of the SN ratio is evaluated.

Original languageEnglish
Pages (from-to)218-229
Number of pages12
JournalJournal of Japan Industrial Management Association
Issue number3
Publication statusPublished - 2015



  • Non-central χ distribution
  • Patnaik approximations
  • Paulson-Takeuchi approximation
  • SN ratio

ASJC Scopus subject areas

  • Industrial and Manufacturing Engineering
  • Applied Mathematics
  • Management Science and Operations Research
  • Strategy and Management

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