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{{Short description|Statistics concept}}
{{More footnotes|date=January 2017}}
{{DISPLAYTITLE:Generalized ''p''-value}}
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In order to overcome the shortcomings of the classical ''p''-values, Tsui and Weerahandi<ref name=TW/> extended the classical definition so that one can obtain exact solutions for such problems as the [[Behrens–Fisher problem]] and testing variance components. This is accomplished by allowing test variables to depend on observable random vectors as well as their observed values, as in the Bayesian treatment of the problem, but without having to treat constant parameters as random variables.
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To describe the idea of generalized ''p''-values in a simple example, consider a situation of sampling from a normal population with the mean <math>\mu</math>, and the variance <math>\sigma ^2</math>. Let <math>\overline{X}</math> and <math>S ^2</math> be the sample mean and the sample variance. Inferences on all unknown parameters can be based on the distributional results
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*Gamage J, Mathew T, and Weerahandi S. (2013). Generalized prediction intervals for BLUPs in mixed models, Journal of Multivariate Analysis}, 220, 226-233.
*Hamada, M., and Weerahandi, S. (2000). Measurement System Assessment via Generalized Inference. Journal of Quality Technology, 32, 241-253.
*Krishnamoorthy, K. and Tian, L. (2007), “Inferences on the ratio of means of two inverse Gaussian distributions: the generalized variable approach”, Journal of Statistical Planning and Inferences, Volume 138, Issue 7, 1
*Li, X., Wang J., Liang H. (2011). Comparison of several means: a fiducial based approach. Computational Statistics and Data Analysis, 55, 1993-2002.
* Mathew, T. and Webb, D. W. (2005). Generalized p-values and confidence intervals for variance components: Applications to Army test and evaluation, Technometrics, 47, 312-322.
*Wu, J. and Hamada, M. S. (2009) Experiments: Planning, Analysis, and Optimization. Wiley, Hoboken, New Jersey.
*Zhou, L., and Mathew, T. (1994). Some Tests for Variance Components Using Generalized p-Values, Technometrics, 36, 394-421.
*Tian, L. and Wu, Jianrong (2006) “Inferences on the Common Mean of
*Tsui, K. and Weerahandi, S. (1989): [https://www.jstor.org/stable/2289949 "Generalized ''p''-values in significance testing of hypotheses in the presence of nuisance parameters"]. ''[[Journal of the American Statistical Association]]'', 84, 602–607
*Weerahandi, S. (1995) [https://www.springer.com/statistics/statistical+theory+and+methods/book/978-0-387-40621-3 ''Exact Statistical Methods for Data Analysis'' ] Springer-Verlag, New York. {{ISBN|978-0-387-40621-3}}
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