Simplified algebraic estimation for the quality control of DIA estimator

Abstract

Based on the unifying framework of the detection, identification and adaption (DIA) estimators, quality control indices are refined and formulated by taking the uncertainty of the combined estimation-testing procedure into account and performing the propagation of uncertainty. These indices are used to measure the confidence levels of the testing decisions, the reliability of the specified alternative hypothesis models, as well as the biasedness and dispersion of the estimated parameters. A simplified algebraic estimation (SAE) method is developed to calculate these quality control indices for the application of single outlier DIA. Compared to the conventional Monte Carlo simulation method, the proposed SAE method can achieve an adequate estimation accuracy and significantly higher computation efficiency. Using a GNSS single-point positioning example, the performance of the SAE method is evaluated and the quality control of the conventionally used DIA estimator is demonstrated for practical applications.

Appendix

The detailed proof for Eqs. (37–39) is as follows.

Under hypothesis ${\mathcal{H}}_{i}$, there is.

$$ w_{i} \sim\{ N\} (\lambda _{i} ,1),w_{j} \sim\{ N\} (\lambda _{i} \rho _{{ji}} ,1),w_{k} \sim\{ N\} (\lambda _{i} \rho _{{jk}} ,1)t)$$

(50)

From Eq. (37), it is derived

$$E\left(\Delta {w}_{jk|i}\right)=\left\{\begin{array}{l}\frac{E\left({w}_{j}-{w}_{k}\right)}{\sqrt{2\left(1-{\rho }_{jk}\right)}}={\lambda }_{i}\frac{\left({\rho }_{ji}-{\rho }_{ki}\right)}{\sqrt{2\left(1-{\rho }_{jk}\right)}}, {\rho }_{ji}\ge 0\;\&\;{\rho }_{ki}\ge 0\\ \frac{E\left({w}_{j}+{w}_{k}\right)}{\sqrt{2\left(1+{\rho }_{jk}\right)}}={\lambda }_{i}\frac{\left({\rho }_{ji}+{\rho }_{ki}\right)}{\sqrt{2\left(1+{\rho }_{jk}\right)}}, {\rho }_{ji}\ge 0\;\&\;{\rho }_{ki}<0\\ -\frac{E\left({w}_{j}+{w}_{k}\right)}{\sqrt{2\left(1+{\rho }_{jk}\right)}}=-{\lambda }_{i}\frac{\left({\rho }_{ji}+{\rho }_{ki}\right)}{\sqrt{2\left(1+{\rho }_{jk}\right)}}, {\rho }_{ji}<0\;\&\;{\rho }_{ki}\ge 0\\ -\frac{E\left({w}_{j}-{w}_{k}\right)}{\sqrt{2\left(1-{\rho }_{jk}\right)}}=-{\lambda }_{i}\frac{\left({\rho }_{ji}-{\rho }_{ki}\right)}{\sqrt{2\left(1-{\rho }_{jk}\right)}}, {\rho }_{ji}<0\;\&\;{\rho }_{ki}<0\end{array}\right.$$

(51)

It can be simplified as

$$E\left(\Delta {w}_{jk|i}\right)=\left\{\begin{array}{l}{\lambda }_{i}\frac{\left(\left|{\rho }_{ji}\right|-\left|{\rho }_{ki}\right|\right)}{\sqrt{2\left(1-{\rho }_{jk}\right)}}, {\rho }_{ji} {\rho }_{ki}\ge 0 \\ {\lambda }_{i}\frac{\left(\left|{\rho }_{ji}\right|-\left|{\rho }_{ki}\right|\right)}{\sqrt{2\left(1+{\rho }_{jk}\right)}}, {\rho }_{ji} {\rho }_{ki}\le 0\end{array}\right.$$

(52)

Similarly, the variance is

$$D\left(\Delta {w}_{jk|i}\right)=\left\{\begin{array}{l}\frac{D\left({w}_{j}\right)+D\left({w}_{k}\right)-2{\rho }_{jk}}{2\left(1-{\rho }_{jk}\right)}=1, {\rho }_{ji}\ge 0 \;\&\;{\rho }_{ki}\ge 0\\ \frac{D\left({w}_{j}\right)+D\left({w}_{k}\right)+2{\rho }_{jk}}{2\left(1+{\rho }_{jk}\right)}=1, {\rho }_{ji}\ge 0\;\&\;{\rho }_{ki}<0\\ \frac{D\left({w}_{j}\right)+D\left({w}_{k}\right)+2{\rho }_{jk}}{2\left(1+{\rho }_{jk}\right)}=1, {\rho }_{ji}<0\;\&\;{\rho }_{ki}\ge 0\\ \frac{D\left({w}_{j}\right)+D\left({w}_{k}\right)-2{\rho }_{jk}}{2\left(1-{\rho }_{jk}\right)}=1, {\rho }_{ji}<0 \;\&\;{\rho }_{ki}<0\end{array}\right.$$

(53)

Therefore, Eqs. (38, 39) are proved.