# Ambiguity-Resolved Model Tests for Carrier-Phase GNSS
## 4 The Ambiguity-Resolved Tests
### AR Identification  
In case \(\mathcal{R}(A,B,C)\) is a strict subspace of \(\mathbb{R}^{m}\), then \(\underline{\xi}_{a}(a)\) also contributes to the statistic (4), thus implying that integer ambiguity estimation under both \(\mathcal{H}_{0}\) and \(\mathcal{H}_{a}\) is required. We then have, with the help of Lemma 1 and the short-hand notation \(\underline{\xi}_{0}=\underline{\xi}_{0}(\underline{\hat{a}}_{0})=\underline{ \hat{a}}_{0}-\underline{\hat{a}}_{0}\) and \(\underline{\xi}_{a}=\underline{\xi}_{a}(\underline{\hat{a}})=\underline{\hat {a}}-\underline{\hat{a}}\), the following representation for the test statistic:  
\[\underline{T}(\underline{\hat{a}}_{0},\underline{\hat{a}})=||\underline{\xi} ||^{2}_{Q_{\text{dr}}}+||\underline{\xi}_{0}||^{2}_{Q_{\text{dr}}\phi_{0}}-|| \underline{\xi}_{a}||^{2}_{Q_{\text{dr}}} \tag{16}\]  
Comparison with the AF statistic \(\underline{T}(\underline{\hat{a}}_{0},\underline{\hat{a}})\) (cf. (6)) shows that now, next to the difference \(||\underline{\xi}||^{2}_{Q_{\text{dr}}}=||\underline{\xi}_{0}||^{2}_{Q_{\text {dr}}}-||\underline{\xi}_{a}||^{2}_{Q_{\text{dr}}}\), the ambiguity-resolution based difference \(||\underline{\xi}_{0}||^{2}_{Q_{\text{dr}}\phi_{0}}-||\underline{\xi}_{a}||^ {2}_{Q_{\text{dr}}}\) is present. It is this added difference, diagnosing the impact of ambiguity resolution under \(\mathcal{H}_{0}\) and \(\mathcal{H}_{a}\), that can be expected to improve the power-performance over that of the AF test. To understand what the best possible performance of the test would be, we determine the PDF of (16) for the limiting case that the ambiguity success rate of \(\underline{\hat{a}}\) goes to one, \(\text{P}[\underline{\hat{a}}=a]\uparrow 1\). The following Theorem shows that the limiting distribution is a _generalized_ Chi-squared distribution, i.e., the distribution of the sum of two independent random variables of which the first is Chi-squared distributed and the second is normally distributed. **Theorem 1** (Limit distribution of \(\underline{T}(\underline{\hat{a}}_{0},\underline{\hat{a}})\)): _Let \(\mathsf{E}(\underline{\hat{a}}|\mathcal{H}_{a})=a\) and \(\mathsf{E}(\underline{\hat{a}}_{0}|\mathcal{H}_{a})=a_{0}\). Then, with the ambiguity success rate under \(\mathcal{H}_{a}\) approaching one, the limiting distribution of the test statistic \(\underline{T}(\underline{\hat{a}}_{0},\underline{\hat{a}})\) is a generalized Chi-squared distribution, given as_  
\[\lim_{\text{P}[\underline{\hat{a}}=a]\uparrow 1}\underline{T}(\underline{\hat{a} }_{0},\underline{\hat{a}})\left\{ \begin{array}{l}\mathcal{H}_{0}\\ \mathcal{H}_{0}\end{array} \chi^{2}(q,0)\right.\]  
_with noncentrality parameter \(\lambda=||c+\hat{C}^{+}A(a-a_{0})||^{2}_{Q_{\text{dr}}(a)\setminus\{\text{dr} \}}\) and normal mean \(\mu=||a-a_{0}||^{2}_{Q_{\text{dr}}}\)._  
Proof. : See Appendix A. This result shows that the limiting distribution of (16) is equal to that of the AK test statistic (10), under \(\mathcal{H}_{0}\) and \(\mathcal{H}_{a}\), in case \(a=a_{0}\). The distributions differ, however, under \(\mathcal{H}_{a}\) when \(a\
eq a_{0}\). Thus, it is the possible bias in \(\underline{\hat{a}}_{0}\) under the alternative hypothesis, i.e., \(\mathsf{E}(\underline{\hat{a}}_{0}|\mathcal{H}_{a})\
eq a\), that drives the difference between the two distributions. A further insight into the behavior of the test is obtained when the difference between \(\underline{\hat{a}}_{0}\) and \(\underline{\hat{a}}\) is neglected.