Saddlepoint Expansions in Linear Regression

In this paper sums of independent but not identically distributed m-dimensional vectors are considered. The summands are generated by a random vector multiplied by a deterministic weight matrix which results in a singular covariance matrix. Under general conditions, given separately for the weight matrix and the random vector, saddlepoint approximations to the distribution of the sum are derived. The results are applied to the least squares estimator, the residual sum of squares, and to an F-statistic in linear regression.