# First and second moments of non-linear least-squares estimators

## Peter J. G. Teunissen

### 2006

Byi = - y, V i " with the aid of we find that b# = - 2 m y 2 + 1/2. 2. $i j$ i J

## Highlights

• Despite the fact that almost all functional relations in our geodetic models are non-linear, we predominantly use the ideas, concepts and results from the theory of linear estimation in our geodetic adjustment
• We will compare our results - parts of which are taken from Teunissen and Teunissen and Knickmeyer - with those of
• Where ~ a. stands for the variance of _xa and I[ by [12 = e - 2 biy gij bJy, w i t h gl"j" gjk = k (~l 9 A similar bound will be derived in section 3 for the bias in the non-linear least-squares estimator
• One can always, from a diagnostic point of view, evaluate the partial derivatives at all likely values of y i, in order to infer whether non-linearity is likely to produce a significant bias or not
• The sample point evaluation introduces namely additional errors that are on the average of the same order as the terms already included in the formula
• The structure of the bias formula for LS--estimators is such that it fits within the general framework of linearized IS-estimation and is not difficult to compute in practice