Abstract
Transformations such as \(V = X + Y -\mathrm{ I}\,[X + Y ]\) or \(W =\min\left (\frac{X} {Y },\, \frac{1-X} {1-Y }\right )\) and Sukhatme’s transformation can be used to augment uniform random samples and uniform order statistics, respectively. We discuss the bearing of these facts in testing uniformity, an important issue in the field of combining p-values in meta-analytical syntheses.
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Notes
- 1.
Observe that if ω X < ∞, we can consider n + 1 spacings, with \(S_{n+1} =\omega _{X} - X_{n:n}\); of course in this situation \(S_{n+1},\,S_{n+1:n+1}\) and W n + 1 (where in this case it is convenient to use the transformation
$$W_{k} = (n + 2 - k)(S_{k:n+1} - S_{k-1:n+1}),$$as in Johnson et al. [8], p. 305) can be expressed as simple functions of the predecessor members of the sequence. We still get the result that \((Y _{1},Y _{2},\ldots ,Y _{n})\mathop{ =}\limits^{ d}(X_{1:n},X_{2:n},\ldots ,X_{n:n})\) in case of standard uniform parent X.
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Acknowledgements
This research has been supported by National Funds through FCT—Fundação para a Ciência e a Tecnologia, project PEst-OE/MAT/UI0006/2011. The authors are grateful to the referees for stimulating comments, leading to further results.
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Brilhante, M.F., Malva, M., Mendonça, S., Pestana, D., Sequeira, F., Velosa, S. (2013). Uniformity. In: Lita da Silva, J., Caeiro, F., Natário, I., Braumann, C. (eds) Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications. Studies in Theoretical and Applied Statistics(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34904-1_7
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