The Modified Best Quadratic Unbiased Non-Negative Estimator (MBQUNE) of variance components

Abstract

Estimated variance components may come out as negative numbers without physical meaning. One way out of this problem is to use non-negative methods. Different approaches have been presented for the solution. Sjoberg presented a method of Best Quadratic Unbiased Non-Negative Estimator (BQUNE) in the Gauss-Helmert model. This estimator does not exist in the general case. Here we present the Modified BQUNE (MBQUNE) obtained by a simple transformation from the misclosures used in the BQUE to residuals. In the Gauss-Markov adjustment model the BQUNE and MBQUNE are identical, and they differ in condition and Gauss-Helmert models only by a simple transformation. If the observations are composed of independent/disjunctive groups the MBQUNE exists in any adjustment model and it carries all the properties of the BQUNE (when it exists). The presented variance component models are tested numerically in some simple examples. It is shown that the MBQUNE works well for disjunctive groups of observations.