You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
In this paper, we study the existence of locally most powerful invariant tests (LMPIT) for the problem of testing the covariance structure of a set of Gaussian random vectors. The LMPIT is the optimal test for the case of close hypotheses, among those satisfying the invariances of the problem, and in practical scenarios can provide better performance than the typically used generalized likelihood ratio test (GLRT). The derivation of the LMPIT usually requires one to find the maximal invariant statistic for the detection problem and then derive its distribution under both hypotheses, which in general is a rather involved procedure. As an alternative, Wijsman's theorem provides the ratio of the maximal invariant densities without even finding an explicit expression for the maximal invariant. We first consider the problem of testing whether a set of N-dimensional Gaussian random vectors are uncorrelated or not, and show that the LMPIT is given by the Frobenius norm of the sample coherence matrix. Second, we study the case in which the vectors under the null hypothesis are uncorrelated and identically distributed, that is, the sphericity test for Gaussian vectors, for which we show that the LMPIT is given by the Frobenius norm of a normalized version of the sample covariance matrix. Finally, some numerical examples illustrate the performance of the proposed tests, which provide better results than their GLRT counterparts.

Cited Abstract(s):
[1]: We consider the hypothesis testing problem of deciding whether an observed high-dimensional vector has independent normal components or, alternatively, if it has a small subset of correlated components. The correlated components may have a certain combinatorial structure known to the statistician. We establish upper and lower bounds for the worst-case (minimax) risk in terms of the size of the correlated subset, the level of correlation, and the structure of the class of possibly correlated sets. We show that some simple tests have near-optimal performance in many cases, while the generalized likelihood ratio test is suboptimal in some important cases.

[2]: The detection and estimation of signals in noisy, limited data is a problem of interest to many scientific and engineering communities. We present a mathematically justifiable, computationally simple, sample-eigenvalue-based procedure for estimating the number of high-dimensional signals in white noise using relatively few samples. The main motivation for considering a sample-eigenvalue-based scheme is the computational simplicity and the robustness to eigenvector modelling errors which can adversely impact the performance of estimators that exploit information in the sample eigenvectors. There is, however, a price we pay by discarding the information in the sample eigenvectors; we highlight a fundamental asymptotic limit of sample-eigenvalue-based detection of weak or closely spaced high-dimensional signals from a limited sample size. This motivates our heuristic definition of the effective number of identifiable signals which is equal to the number of ldquosignalrdquo eigenvalues of the population covariance matrix which exceed the noise variance by a factor strictly greater than . The fundamental asymptotic limit brings into sharp focus why, when there are too few samples available so that the effective number of signals is less than the actual number of signals, underestimation of the model order is unavoidable (in an asymptotic sense) when using any sample-eigenvalue-based detection scheme, including the one proposed herein. The analysis reveals why adding more sensors can only exacerbate the situation. Numerical simulations are used to demonstrate that the proposed estimator, like Wax and Kailath's MDL-based estimator, consistently estimates the true number of signals in the dimension fixed, large sample size limit and the effective number of identifiable signals, unlike Wax and Kailath's MDL-based estimator, in the large dimension, (relatively) large sample size limit.

[3]: A problem occurring in radio astronomy is the detection and cancellation of spatially correlated interfering signals entering via the sidelobes of the telescopes in an array. A complicating factor is that the noise powers can be different at each telescope. For the case that the sensors are uncalibrated, we formulate the detection problem as a test on the covariance structure, state the GLRT for this problem, and relate it to a simpler ad-hoc detector. We derive algorithms to estimate the noise powers and the subspace of interferer signature vectors. Once the subspace is estimated, the interference can be projected out. We compare this method to the conventional multichannel subspace detector and show its robustness to non-identical channels on data collected with the Westerbork radio telescope.

[4]: We consider the detection of unknown Gaussian signals received by an array of uncalibrated nonidentical sensors, which is a problem that appears in radio astronomy. The problem is formulated as a test on the covariance structure. The generalized likelihood ratio test (GLRT) for this problem is stated and related to a simpler ad-hoc detector. We compare the method to the conventional multichannel subspace detector and show its robustness to nonidentical channels on data collected with the Westerbork radio telescope.
Correlation Test : For vectors with different covariances under the null hypothesis, we show that the LMPIT is given by the Frobenius norm of the sample coherence matrix. The block diagram of the LMPIT is depicted in the lower part of Figure , where we can see that the whitening is different for each vector. We should also point out that the Frobenius norm of the coherence matrix, or similar squared-sum test statistics, have been previously proposed as approximations of the GLRT that might be computationally simpler or have some other advantage [3] [4] [1] [2] . Here, we prove for the first time that the Frobenius norm of the coherence matrix is in fact the LMPIT for this problem.