Biometrika Advance Access originally published online on January 31, 2008
Biometrika 2008 95(1):1-16; doi:10.1093/biomet/asm093
Articles |
Studentization and deriving accurate p-values
Department of Statistics, University of Toronto, Toronto, M5S 3G3, Canada dfraser{at}utstat.toronto.edu
CEREMADE, University Paris Dauphine, 75016 Paris, France rousseau{at}ceremade.dauphine.fr
Received for publication 1 January 2005. Revision received 1 August 2007.
We have a statistic for assessing an observed data point relative to a statistical model but find that its distribution function depends on the parameter. To obtain the corresponding p-value, we require the minimally modified statistic that is ancillary; this process is called Studentization. We use recent likelihood theory to develop a maximal third-order ancillary; this gives immediately a candidate Studentized statistic. We show that the corresponding p-value is higher-order Un(0, 1), is equivalent to a repeated bootstrap version of the initial statistic and agrees with a special Bayesian modification of the original statistic. More importantly, the modified statistic and p-value are available by Markov chain Monte Carlo simulations and, in some cases, by higher-order approximation methods. Examples, including the Behrens–Fisher problem, are given to indicate the ease and flexibility of the approach.
Key Words: Ancillary • Bayesian • Behrens–Fisher problem • Bootstrap • Conditioning • Departure measure • Likelihood • p-value • Studentization
References
-
Barndorff-Nielsen O. E. Inference on full or partial parameters based on the standardised signed log likelihood ratio. Biometrika (1986) 73:307–22.
Bayarri M. J., Berger J. O. p-values for composite null models. J. Am. Statist. Assoc. (2000) 95:1127–42.[CrossRef][Web of Science]
Behrens W. V. Ein Beitrag zur Fehlerbereichnung bei wenigen Beobachtungen. Landwirt. Jahr. (1929) 68:807–37.
Beran R. J. Pre-pivoting test statistics: A bootstrap view of asymptotic refinements. J. Am. Statist. Assoc. (1988) 83:687–97.[CrossRef][Web of Science]
Bhattacharya R. N., Ghosh J. K. On the validity of the formal Edgeworth expansion. Ann. Statist. (1978) 6:434–51.[CrossRef]
Bickel P. J., Ghosh J. K. A decomposition for the likelihood ratio statistic and the Bartlett correction—A Bayesian argument. Ann. Statist. (1990) 18:1070–90.[CrossRef]
Box G. E. P. Sampling and Bayes inference in scientific modelling (with Discussion). J. R. Statist. Soc. A (1980) 143:383–430.
Cakmak S., Fraser D. A. S., McDunnough P., Reid N., Yuan X. Likelihood centered asymptotic model: Exponential and location model versions. J. Statist. Plan. Infer. (1998) 66:211–22.[CrossRef]
Cakmak S., Fraser D. A. S., Reid N. Multivariate asymptotic model: Exponential and location approximations. Utilitas Math. (1994) 46:21–31.
Davison A. C., Fraser D. A. S., Reid N. Improved likelihood inference for discrete data. J. R. Statist. Soc. B (2006) 68:495–508.[CrossRef]
Fisher R. A. The logic of inductive inference. J. R. Statist. Soc. (1935) 98:39–54.[CrossRef]
Fraser D. A. S. Likelihood for component parameters. Biometrika (2003) 90:327–39.
Fraser D. A. S. Ancillaries and conditional inference (with Discussion). Statist. Sci. (2004) 19:333–69.[CrossRef]
Fraser D. A. S., Reid N. Third order asymptotic models: Likelihood functions leading to accurate approximations for distribution functions. In: Statist. Sinica (1993) 3:67–82.
Fraser D. A. S., Reid N. Ancillaries and third order significance. Utilitas Math. (1995) 47:330–53.
Fraser D. A. S., Reid N. Ancillary information for statistical inference. Empirical Bayes and Likelihood Inference—Ahmed S. E., Reid N., eds. (2001) New York: Springer-Verlag. 185–207.
Fraser D. A. S., Reid N., Wu J. A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika (1999) 86:249–64.
Gelman A., Carlin J. B., Stern H., Rubin D. B. Bayesian Data Analysis. (1995) London: Chapman and Hall.
Ghosh M., Kim Y.-H. The Behrens–Fisher problem revisited: A Bayesian-frequentist synthesis. Can. J. Statist. (2001) 29:5–17.[CrossRef]
Jeffreys H. Theory of Probability (1961) 3rd ed. Oxford: Oxford University Press.
Lugannani R., Rice S. O. Saddlepoint approximation for the distribution of the sums of independent random variables. Adv. Appl. Prob. (1980) 12:457–90.
Robins J. M., van der Vaart A., Ventura V. The asymptotic distribution of p-values in composite null models. J. Am. Statist. Assoc. (2000) 95:1143–56.[CrossRef][Web of Science]
Rousseau J. Approximating interval hypotheses : p-values and Bayes factor. In: Bayesian Statistics—Bernardo J. M., Berger J. O., Dawid A. P., Smith A. F. M., eds. (2007) 8. Oxford: Oxford University Press. 417–52.
Student. The probable error of a mean. Biometrika (1908) 6:1–25.
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