# Ambiguity-Resolved Model Tests for Carrier-Phase GNSS
## 1 Introduction
We also show, when the mixed-integer misclosure vector is used to formulate a derived simple null hypothesis, how its distribution can be used to formulate a highest-density (HD) detection test. This test will have similar performance but is somewhat more difficult to execute. We show an illustrative example in which the power of the MIM test, although not as good as the AK test, is still significantly better than the AF test. In Section 4.2, we discuss the MIM test for the case of identification, i.e., when the design matrix' range space of the alternative hypothesis is a strict subspace of the observation space. In this case, integer ambiguity resolution is required under both the null and alternative hypotheses. We prove that the test statistic has a _generalized_ Chi-square distribution if the ambiguity success rate approaches one under the alternative hypothesis. A further insight into the test is provided by neglecting the ambiguity resolution difference of the two hypotheses. This results in a significance test for which the distribution of the test statistic is also given. Again, we also provide the associated HD significance test. For the cases in which _full_ ambiguity resolution is not sufficient for a significant power improvement, we show how _partial_ ambiguity resolution under the alternative hypothesis may drastically improve the power performance. In Section 4.3, we describe a third important application of the MIM test. This concerns the case under which the ambiguities of the null hypothesis are assumed integer and those under the alternative hypothesis as real-valued. This form of the MIM test is of interest for problems that require _integer testing_ of the ambiguities. Such problems are also of interest for carrier-phase GNSS, since the stability of instrumental hardware delays may not always be such that integers of the carrier-phase ambiguities is guaranteed. In [14; 15], for instance, it is demonstrated that smartphone phase observations may be contaminated by receiver effects that destroy the integer property of the ambiguities. We describe the integer test, its associated distributions, and provide an illustrative example of its power performance. Finally, we provide a summary and conclusions in Section 5. The following notation is used throughout. We denote a random variable/vector by means of an underscore; thus, \(\underline{x}\) is a random variable/vector, while \(x\) is not. \(\mathsf{E}(\underline{x})\) and \(\mathsf{D}(\underline{x})\) stand for the expectation and dispersion of \(\underline{x}\), respectively, and \(\mathcal{N}_{p}(\mu,Q)\) denotes a \(p\)-dimensional, normally distributed random vector, with mean (expectation) \(\mu\) and variance matrix (dispersion) \(Q\). The Best Linear Unbiased Estimator (BLUE) of a parameter vector \(x\) is denoted as \(\underline{x}\) and its admissible integer estimator as \(\underline{x}\). \(\mathsf{P}[\mathcal{A}]\) denotes the probability of event \(\mathcal{A}\), \(f_{t}(x)\) the probability density function (PDF) of the continuous random vector \(\underline{x}\), and \(\mathsf{P}[\underline{a}=z]\) the probability mass function (PMF) of the integer random vector \(\underline{a}\). The noncentral Chi-square distribution with \(q\) degrees of freedom and noncentrality parameter \(\lambda\) is denoted as \(\chi^{2}(q,\lambda)\). The range space of a matrix \(M\) is denoted as \(\mathcal{R}(M)\), and \(\mathbb{R}^{p}\) and \(\mathbb{Z}^{p}\) denote the \(p\)-dimensional spaces of real- and integer numbers, respectively. The \(Q_{yy}\)-weighted squared norm is denoted as \(||.||^{2}_{Q_{yy}}=(.)^{T}Q_{yy}^{-1}(.)\). The BLUE-inverse of a full column rank matrix \(M\) is denoted as \(M^{+}=(M^{T}Q_{yy}^{-1}M)^{-1}M^{T}Q_{yy}^{-1}\) and the orthogonal projector onto \(\mathcal{R}(M)\) as \(P_{M}=MM^{+}\). Thus, \(P_{M}^{\perp}=I-P_{M}\) is the orthogonal projector that projects orthogonally on the orthogonal complement of \(\mathcal{R}(M)\).