By Van Der Merwe A. J., Du Plessis J. L.
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Additional resources for A Bayesian Approach to Selection and Ranking Procedures: The Unequal Variance Case
F. f. of the t distribution with v = n — 2 degrees of freedom. f. of the studentized deleted residuals after the observations were weighted. f. f. of the t distribution with v = n - 2 = 45. We now need to calculate a test statistic to test H0 : B1 = 0 versus Ha : B1 0. Under the null hypothesis H0 :b1 = 0 the model becomes This model, which was used to calculate the weights, will be called the reduced model. We can use WLS to minimize the weighted sum of squared errors for the reduced model as This minimum value will be indicated by SSE*R.
However, if the errors are not normally distributed, then the F test may not be the most powerful test. We turn now to an adaptive test that is also robust for validity and often has greater power than the F test if the errors are not normal. 2 Computing and Smoothing Residuals For this general adaptive test we will use the studentized deleted residuals to weight the observations in the same way as they were used in the two-sample test. For the general linear model Belsley, Kuh, and Welsch (1980) express the studentized deleted residuals for the ith observation in the reduced model as where ei is the ordinary residual, hii is the ith diagonal element of the hat matrix X R ( X ' R X R ) - l X ' R , and s(i) is the estimate of based on the n — 1 observations obtained by deleting the ith observation from the data set.
F. f. of the t distribution, then the observations will be weighted so that, after weighting, the distribution of the studentized deleted residuals will more closely approximate the t distribution. f. 4. Weighting Observations 21 Tn-2 (•), and the smaller residuals need to be increased in size. This is accomplished by giving weights less than one to the observations with the largest residuals and greater than one to observations with the most highly negative residuals. The rationale for using the weights wi, = ti,/