Computation of a Test Statistic in Data Quality Control

When processing observational data, statistical testing is an essential instrument to hopefully render harmless incidental anomalies and disturbances in the measurements. A commonly used test statistic based on the general linear model is the generalized likelihood ratio test statistic. The standard formula given in the literature for this test statistic is not defined if the noise covariance matrix is singular, and is not suitable for computation if any of the matrices involved is ill-conditioned. Based on Paige’s generalized linear least squares method [Comm. Statist., Vol. B7, 437–453, 1978], a numerically stable approach is proposed for the computation of the test statistic, as well as for the estimates of the parameter vectors, and reliable representations of the error covariance matrices for these estimates are presented. This approach allows the noise covariance matrix to be singular, and can be applied directly to the linear model with linear equality constraints.