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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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