Saddlepoint approximation and bootstrap inference for the Satterthwaite class of ratios

Saddlepoint approximations are developed for the distributions of Satterthwaite F-type statistics that arise in ANOVA with balanced random-effects models. Such statistics are commonly used in variance component testing and confidence interval construction. because exact F pivotals usually do not exist. The approximations are Shown to be uniform in their right tails, and the limiting relative errors for distribution approximations are determined. Based on these approximations, a new method is devised for confidence interval construction of variance component ratios. Simulations show that the method has superior coverage accuracy over existing methods. Prepivoting of the proposed method, using the double-parametric bootstrap, is implemented and shows further coverage improvement. The double bootstrap is most efficiently implemented by using the saddlepoint approximation in lieu of the inner layer of resampling: therefore, only it single outer layer of resampling is required.