# Ambiguity-Resolved Model Tests for Carrier-Phase GNSS
## Appendix A Proof of Lemma 1 (Differences normed residuals). Using the projector product formula, \(P_{[A,B,C]}^{\perp}=P_{\mathcal{C}}^{\perp}P_{[A,B]}^{\perp}\), with \(\tilde{C}=P_{[A,B]}^{\perp}\tilde{C}\), we may express the \(\mathcal{H}_{a}\)-residual vector \(\underline{\hat{z}}_{a}=P_{[A,B,C]}^{\perp}\underline{y}\) in the \(\mathcal{H}_{0}\)-residual vector \(\underline{\hat{z}}_{0}=P_{[A,B]}^{\perp}\underline{y}\) as \(\underline{\hat{z}}_{a}=P_{\tilde{C}}^{\perp}\underline{\hat{z}}_{0}\). This, together with \begin{table}
\begin{tabular}{l l l l} \hline \hline  & **Ambiguity-Float (AF)** & **Ambiguity-Known (AK)** & **Ambiguity-Resolved (AR)** \\ \hline \multirow{3}{*}{Detection} & \(T(a_{0},\bullet)=||a_{0}||_{Qay}^{2}\) & \(T(a,\bullet)=||a_{0}||_{Qay}^{2}+||a_{0}(a)||_{Qay}^{2}\) & \(T(a_{0},\bullet)=||a_{0}||_{Qay}^{2}+||a_{0}(a_{0})||_{Qay}^{2}\) \\  & \(\mathcal{R}(A,B,C)=\mathbb{R}^{m}\) & \(\mathcal{R}(B,C)=\mathbb{R}^{m}\) & \(\mathcal{R}(A,B,C)=\mathbb{R}^{m}\) \\ \hline \multirow{3}{*}{Identification} & \(T(a_{0},\underline{A})=||e||_{Qay}^{2}\) & \(T(a,a)=||e(a)||_{Qay}^{2}\) & \(T(a_{0},\underline{A})=||e(a)||_{Qay}^{2}+||e_{Qay}(a)||_{Qay}^{2}+||e_{Qay}(a)||_{Qay}^{ 2}\) \\  & & \(||e_{0}(a_{0})||_{Qay}^{2}-||e_{0}(a)||_{Qay}^{2}\) & \(||e_{0}(a)||_{Qay}^{2}\) \\ \hline Integer Testing & & \(T(a_{0},\underline{A})=||e||_{Qay}^{2}+||e_{0}(a_{0})||_{Qay}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mixed-integer model (MIM) testing: an overview of the AF, AK, and AR test statistics. \(P_{\mathcal{C}}=\tilde{\mathcal{C}}\tilde{\mathcal{C}}^{+}\) and the AF-BLUE of \(c\) expressed as \(\underline{\hat{c}}=\tilde{\mathcal{C}}^{+}\underline{y}=\tilde{\mathcal{C}}^{+ }\underline{\hat{c}}_{0}\), having the variance matrix \(Q_{\mathcal{C}\varepsilon}=(\tilde{\mathcal{C}}^{T}Q_{gy}^{-1}\tilde{\mathcal{ C}})^{-1}\), allows us to write  
\[ \begin{array}{rcl}||\hat{\varrho}_{0}||_{Q_{gy}}^{2}&=&||P_{\mathcal{C}}^{ \perp}\underline{\hat{c}}_{0}||_{Q_{gy}}^{2}+||P_{\mathcal{C}}\underline{\hat {c}}_{0}||_{Q_{gy}}^{2}\\ &=&||\underline{\hat{c}}_{a}||_{Q_{gy}}^{2}+||\tilde{\mathcal{C}}\tilde{ \mathcal{C}}^{+}\underline{\hat{c}}_{0}||_{Q_{gy}}^{2}\\ &=&||\underline{\hat{c}}_{a}||_{Q_{gy}}^{2}+||\underline{\hat{c}}||_{Q_{cx}}^{2 }\end{array} \]  
which proves Lemma 1(_i_). The resulting Lemma 1(_ii_), in which \(\underline{\hat{c}}(a)\) is the AK-BLUE of \(c\), is proven in a similar way. To prove Lemma 1(_iii_), we use the projector product formula \(P_{[A,B]}^{\perp}=P_{A}^{\perp}P_{B}^{\perp}\), with \(\bar{A}=P_{B}^{\perp}A\), to express the AF residual vector \(\underline{\hat{c}}_{0}=P_{[A,B]}^{\perp}(\underline{y}-Aa)\) in the AK residual vector \(\underline{\hat{c}}_{0}(a)=P_{B}^{\perp}(\underline{y}-Aa)\) as \(\underline{\hat{c}}_{0}=P_{A}^{\perp}\underline{\hat{c}}_{0}(a)\). This, together with \(P_{A}=\bar{A}\bar{A}^{+}\), \(\underline{\hat{a}}_{0}-a=\bar{A}^{+}(\underline{y}-Aa_{0})\), and \(Q_{abab}=(\bar{A}^{T}Q_{gy}^{-1}\bar{A})^{-1}\), allows us to write  
\[ \begin{array}{rcl}||\underline{\hat{c}}_{0}(a)||_{Q_{gy}}^{2}&=&||P_{A}^{ \perp}\underline{\hat{c}}_{0}||_{Q_{gy}}^{2}+||P_{A}\underline{\hat{c}}_{0}(a )||_{Q_{gy}}^{2}\\ &=&||\underline{\hat{c}}_{0}||_{Q_{gy}}^{2}+||\bar{A}\bar{A}^{+}(\underline{y} -Aa)||_{Q_{gy}}^{2}\\ &=&||\underline{\hat{c}}_{0}||_{Q_{gy}}^{2}+||\underline{\hat{c}}_{0}-a||_{Q_{ y}0}^{2}\end{array} \]  
which proves Lemma 1(_iii_). The resulting Lemma 1(_iv_) is proven in a similar way. Proof of Theorem 1 (Limit distribution of \(\underline{T}(\underline{\hat{a}}_{0},\underline{\hat{a}})\)). : As \(\underline{\hat{a}}\) and \(\underline{\hat{a}}_{0}\) are both unbiased estimators under \(\mathcal{H}_{0}\), we have, since--\(Q_{\underline{\hat{a}}\hat{a}}\geq Q_{\hat{a}_{0}\hat{a}_{0}}\)--that under \(\mathcal{H}_{0}\), \(\underline{\hat{a}}\to a\) and \(\underline{\hat{a}}_{0}\to a\), when \(\mathsf{P}[\underline{\hat{a}}=a]\to 1\).