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Biometrika 1990 77(2):333-341; doi:10.1093/biomet/77.2.333
© 1990 by Biometrika Trust
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Nonnormal linear regression; An example of significance levels in high dimensions

D. A. S. FRASER, H. S. LEE and N. REID

Department of Mathematics, York University Downsview, Ontario, Canada M3J 1P3
Department of Statistics, University of Toronto Toronto, Canada M5S 1A1

Analysis of nonnormal linear models leads to an initial conditioning on the standardized residuals, giving an unnormed density on Rk, where k is the number of parameters. To obtain an observed level of significance for a single parameter it is then necessary to calculate a marginal probability, thus requiring integration in k dimensions. In this paper a conditional approach to evaluating the observed level of significance is developed, and an importance sampling technique is used to improve the approximation and assess the accuracy of the conditional approximation to the marginal observed level. A further approximation based on the invariant version (Fraser, 1990) of the Lugannam & Rice (1980) formula is also proposed. The approach extends to the evaluation of real pivots and to Bayesian inference for a single parameter component.

Key Words: Bayesian analysis • Conditional inference • Linear model • Nonnormal analysis • Observed level of significance • Regression model


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