Adjustment of geodetic measurements with mixed multiplicative and additive random errors

Comparing the estimates of the variance of unit weight in multiplicative error models

“…For more details, the reader is referred to Xu and Shimada (2000), Xu et al (2013) and Shi et al (2014).…”

“…whereβ W L S is the weighted LS estimator of β. Sinceβ W L S is known to be biased (see also Xu and Shimada 2000;Xu et al 2013), it does not possess the optimal statistical properties of unbiasedness and minimum variance, as in the case of linear models with additive random errors. It is also well known from Xu and Shimada (2000) and Xu et al (2013) that the bias of the weighted LS estimateβ WLS is completely attributed to the dependence of the variancecovariance matrix Σ y of the measurements on the unknown parameters β.…”

“…In fact, electromagnetic-wave-based geodetic measurements have unambiguously demonstrated by themselves that measurements of these types are contaminated by both additive and multiplicative random errors (see, e.g. Xu et al 2013;Shi et al 2014). The vector form of a multiplicative error model can be symbolically represented as follows:…”

“…Although multiplicative errors should dominate almost all geodetic measurements of modern types, they have been investigated only recently in geodesy (see, e.g. Xu et al 2013;Shi et al 2014). Unlike the standard method of quasi-likelihood in statistics, geodesists approach to handle multiplicative error models from the point of view of the conventional least squares (LS) principle (see, e.g.…”

“…Unlike the standard method of quasi-likelihood in statistics, geodesists approach to handle multiplicative error models from the point of view of the conventional least squares (LS) principle (see, e.g. Xu 1999;Xu and Shimada 2000;Xu et al 2013;Shi et al 2014). The advantage of using the LS principle is twofold: (i) we do not assume probability distributions for measurements; and (ii) the LS objective function is well defined.…”