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

Abstract

Multiplicative error models should become more and more important in geodesy, since modern measurement technology on the basis of electromagnetic wave has clearly demonstrated that measurements of this type contain two types of random errors: fixed random errors and baseline-length dependent random errors. Although a number of the estimators of the variance of unit weight are derived from the least-squares-based adjustment methods for multiplicative error models recently, we know very little about their statistical performances. We first derive the variances of the estimates of the variance of unit weight in multiplicative error models. We find that the second order term of random errors will not affect the unbiasedness of an estimate of the variance of unit weight, if such a term is generated from the nonlinearity of models and/or least-squares-based nonlinear objective functions. The result is surprising, since the second order term of random errors has been well known to create the biases in both the estimate of parameters and the measurement corrections in the literature of nonlinear adjustment and nonlinear regression. Simulations are carried out to confirm the statistical analysis and to numerically compare the performances of different estimates of the variance of unit weight in multiplicative error models. From the simulation results, we recommend the estimate of the variance of unit weight with the bias-corrected weighted LS solution, followed by the two estimates with the ordinary LS solutions and the first estimate with the weighted LS solution.

Unstable Object Points during Measurements—Deformation Analysis Based on Pseudo Epoch Approach

“…Here, we consider the Huber method, so the estimator of the parameter vector is determined as follows [ 16 , 35 , 36 ]: where W is the diagonal matrix of weights , is the weight function related to a variant of M-estimation and , is the standardized error of i th observation, and is the i th diagonal element of a matrix. The standardized error can be computed in the following way: where is the standard deviation of the error that can be computed by applying the approximation of the covariance matrix of errors in the following well-known form: where is the variance of unit weight [ 37 ]. Another way to determine the error standard deviation is based on the application of Monte Carlo simulations [ 38 ].…”

Accuracy of the Hodges–Lehmann estimates computed by applying Monte Carlo simulations

“…Thus, we will apply one of the possible approaches to that problem, namely Monte Carlo simulations (other possible ways of assessing the accuracy of median-based estimates can be found in, e.g., Maritz and Jarret 1978;Hettmansperger and McKean 2011). The method which is based on Monte Carlo simulations was already applied for assessing the accuracy of several estimates (e.g., Duchnowski 2013; Duchnowski and Wiśniewski 2014;Shi and Xu 2015) and it seems to be very useful and advisable especially in the case of HLWE (which is more complicated case because of the application of the weighted median). Duchnowski (2013) proposed to assess the accuracy of HLWE in relation to the accuracy of the respective least squares estimates (LSE) which seems very advisable from the practical point of view.…”