EXACT DISTRIBUTIONS OF R^2 AND ADJUSTED R^2 IN A LINEAR REGRESSION MODEL WITH MULTIVARIATE t ERROR TERMS

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In this paper we consider a linear regression model when error terms obey a multivariate t distribution, and examine the effects of departure from normality of error terms on the exact distributions of the coefficient of determination (say, R^2) and adjusted R^2 (say, R^2 ). We derive the exact formulas for the density function, distribution function and m-th moment, and perform numerical analysis based on the exact formulas. It is shown that the upward bias of R^2 gets serious and the standard error of R^2 gets large as the degrees of freedom of the multivariate t error distribution (say, V_0) get small. The confidence intervals of R^2 and R^2 are examined, and it is shown that when the values of V_0 and the parent coefficient of determination (say, Φ) are small, the upper confidence limits are very large, relative to the value of Φ.
日本統計学会の論文