The geometry of estimation and hypothesis testing in the constrained linear model-the full rank case

Abstract

The classical approach to linear models is algebraic in nature, with attention being focused on parameter space. The parameter vector is estimated by minimizing a quadratic form. (In the constrained model this minimization is accomplished via the method of Lagrange Multipliers.) Transforming this estimate with the design matrix yields the estimate of the mean for the observation vector. The numerator sum of squares for testing a null hypothesis in the constrained model is given by the difference between two quadratic forms in the estimate of the parameter vector for the unconstrained model.

The setting for the geometric approach is observation space. The mean (observation) vector is estimated directly by orthogo nally projecting the vector of observations onto an appropriate subspace, whether the. model is constrained or not. The estimate of the parameter vector is obtained from this estimate by the appropriate linear transformation. There is no need to employ the Lagrange Multiplier technique in the constrained model. The geometry of observation space reveals chat the numerator sum of squares for testing the null hypothesis in the constrained model may be expressed as a single quadratic form in the estimate of the parameter vector for the constrained model. This single quadratic form is computationally more efficient than the difference between the two quadratic forms previously mentioned.

Keywords:

• observation space
• parameter space
• orthogonal projection
• orthogonal decomposition
• quad- ratic form
• best linear unbiased estimate
• F statistic