Describing the Quality of Inequality Constrained Estimates

A key feature of geodetic adjustment theory is the description of stochastic properties of the estimated quantities. A variety of tools and measures have been developed to describe the quality of ordinary least-squares estimates, for example, variance-covariance information, redundancy numbers, etc. Many of these features can easily be extended to a constrained least-squares estimate with equality constraints. However, this is not true for inequality constrained estimates. In many applications in geodesy the introduction of inequality constraints could improve the results (e.g. filter and network design or the regularization of ill-posed problems). This calls for an adequate stochastic modeling accompanying the already highly developed estimation theory in the field of inequality constrained estimation. Therefore, in this contribution, an attempt is made to develop measures for the quality of inequality constrained least-squares estimates combining Monte Carlo methods and the theory of quadratic programming. Special emphasis is placed on the derivation of confidence regions.