Systematic bias of selected estimates applied in vertical displacement analysis

Abstract

<jats:title>Abstract</jats:title><jats:p>In surveying problems we almost always use unbiased estimators; however, even unbiased estimator might yield biased assessments, which is due to data. In statistics one distinguishes several types of such biases, for example, sampling, systemic or response biases. Considering surveying observation sets, bias from data might result from systematic or gross errors of measurements. If nonrandom errors in an observation set are known, then bias can easily be determined for linear estimates (e.g., least squares estimates). In the case of non-linear estimators, it is not so simple. In this paper we are focused on a vertical displacement analysis and we consider traditional least squares estimate, two M<jats:italic><jats:sub>split</jats:sub></jats:italic>estimates and two basic robust estimates, namely M-estimate, R-estimate. The main aim of the paper is to assess estimate biases empirically by applying Monte Carlo method. The smallest biases are obtained for M- and R-estimates, especially for a high magnitude of a gross error. On the other hand, there are several cases when M<jats:italic><jats:sub>split</jats:sub></jats:italic>estimates are the best. Such results are acquired when the magnitude of a gross error is moderate or small. The outcomes confirm that bias of M<jats:italic><jats:sub>split</jats:sub></jats:italic>estimates might vary for different point displacements.</jats:p>

Determination of Terrain Profile from TLS Data by Applying Msplit Estimation

“…Applications of M split estimation concerned so far not only processing of TLS data (presented in [1,15,22,31]) but also deformation analyses [29,30,[32][33][34][35][36], direct identification of gross errors [37], linear regression analyses [30,34], robust coordinate transformation [38], S-transformation [39] and marine navigation [40]. Most of these applications applied only the squared M split estimation (SMS), of which the objective functions stem from the assumption that observation errors are normally distributed.…”