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The Rule of Three, its Variants and Extensions

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  • Frank Tuyl
  • Richard Gerlach
  • Kerrie Mengersen

Abstract

The Rule of Three (R3) states that 3/n is an approximate 95% upper limit for the binomial parameter, when there are no events in n trials. This rule is based on the one‐sided Clopper–Pearson exact limit, but it is shown that none of the other popular frequentist methods lead to it. It can be seen as a special case of a Bayesian R3, but it is shown that among common choices for a non‐informative prior, only the Bayes–Laplace and Zellner priors conform with it. R3 has also incorrectly been extended to 3 being a “reasonable” upper limit for the number of events in a future experiment of the same (large) size, when, instead, it applies to the binomial mean. In Bayesian estimation, such a limit should follow from the posterior predictive distribution. This method seems to give more natural results than—though when based on the Bayes–Laplace prior technically converges with—the method of prediction limits, which indicates between 87.5% and 93.75% confidence for this extended R3. These results shed light on R3 in general, suggest an extended Rule of Four for a number of events, provide a unique comparison of Bayesian and frequentist limits, and support the choice of the Bayes–Laplace prior among non‐informative contenders. La Règle de Trois établit que 3/n est une limite supérieure approchée de 95% pour le paramètre de la loi binomiale quand aucun événement n'est observé en n essais. Cette règle est basée sur la limite unilatérale de Clopper‐Pearson, mais il est montré qu'aucune des autres méthodes populaires fréquentistes ne s'y ramène. Elle peut être vue comme un cas particulier d'une Règle Bayésienne de Trois, mais il est montré que, parmi les choix classiques d'a priori non‐informatif, seulement les a priori de Bayes‐Laplace et de Zellner s'y conforment. La Règle de Trois a aussi été inexactement prolongée à 3 étant une limite supérieure ‘raisonnable’ pour le nombre d'événements dans une future expérience de même (grande) taille. En estimation Bayésienne, une telle limite devrait découler de la distribution prédictive a posteriori. Cette méthode semble donner des résultats plus normaux que, bien que lorsqu'elle est basée sur l'a priori de Bayes‐Laplace converge techniquement avec, la méthode de limites fréquentistes de prédiction qui indiquent un niveau de confiance entre 87.5% et 93.75% pour cette Règle prolongée de Trois. Ces résultats amène un nouvel éclairage sur la Règle de Trois en général, suggérent une Règle prolongée de Quatre pour un nombre d'événements, fournissent une comparaison unique des limites Bayésiennes et fréquentistes, et renforcent le choix de l'a priori de Bayes‐Laplace parmi les lois non‐informatives concurrentes.

Suggested Citation

  • Frank Tuyl & Richard Gerlach & Kerrie Mengersen, 2009. "The Rule of Three, its Variants and Extensions," International Statistical Review, International Statistical Institute, vol. 77(2), pages 266-275, August.
  • Handle: RePEc:bla:istatr:v:77:y:2009:i:2:p:266-275
    DOI: 10.1111/j.1751-5823.2009.00078.x
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    References listed on IDEAS

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    1. Tuyl, Frank & Gerlach, Richard & Mengersen, Kerrie, 2008. "A Comparison of BayesLaplace, Jeffreys, and Other Priors: The Case of Zero Events," The American Statistician, American Statistical Association, vol. 62, pages 40-44, February.
    2. Paul H. Kupiec, 1995. "Techniques for verifying the accuracy of risk measurement models," Finance and Economics Discussion Series 95-24, Board of Governors of the Federal Reserve System (U.S.).
    3. Robert F. Engle & Simone Manganelli, 2004. "CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles," Journal of Business & Economic Statistics, American Statistical Association, vol. 22, pages 367-381, October.
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    Cited by:

    1. Jiajin Wei & Ping He & Tiejun Tong, 2024. "Estimating the Reciprocal of a Binomial Proportion," International Statistical Review, International Statistical Institute, vol. 92(1), pages 1-16, April.
    2. Paul H. Garthwaite & John R. Crawford, 2011. "Inference for a binomial proportion in the presence of ties," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(9), pages 1915-1934, October.
    3. Frank Tuyl, 2017. "A Note on Priors for the Multinomial Model," The American Statistician, Taylor & Francis Journals, vol. 71(4), pages 298-301, October.

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