Distribution of some statistics in the presence of blunders in geodetic observations

Abstract

In presence of gross errors (blunders) in the observations, we derive the distributions of some statistics defined after a least-squares (LS) adjustment. Quadratic forms of the LS-residuals and LS-estimates of the blunders are considered along with some their combinations. We demonstrate, in two different ways, that the difference between the quadratic form of the LS-residuals and the quadratic form of the LS-estimates of the blunders has a central chi-square distribution, while each of these two quadratic forms has a non-central chi-square distribution with the same non-centrality parameter. Finally we prove the noncentral Fisher distribution of the ratio between the quadratic form of the LS-estimates of the blunders and the difference of the above-mentioned quadratic forms.