On Best Linear Estimation and General Gauss-Markov Theorem in Linear Models with Arbitrary Nonnegative Covariance Structure

Key Points

• To identify the complete class of conditional inverses for arbitrary nonnegative covariance matrices that yield best linear unbiased estimators in general linear models.
• Formulated general normal equations using a specified class of conditional inverses for known, possibly singular nonnegative covariance matrices.
• Derived algebraic and statistical properties of estimator solutions under general linear and multivariate normal assumptions.
• Established a general hypothesis testing framework based on solutions to the revised normal equations.
• Identified the complete nonempty class of conditional inverses that ensures solutions to general normal equations yield best linear unbiased estimators for any estimable parametric function.
• Demonstrated that the derived estimators correspond directly to maximum likelihood estimators when errors follow a multivariate normal distribution.
• Formulated an exact procedure for linear hypothesis testing directly using solutions derived from the general normal equations.