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Kelly's (1986) generalized likelihood ratio test (GLRT) statistic is reexamined under a broad class of data distributions known as complex multivariate elliptically contoured (MEC), which include the complex Gaussian as a special case. We show that, mathematically, Kelly's GLRT test statistic is again obtained when the data matrix is assumed to be MEC distributed. The maximum-likelihood (ML) estimate for the signal parameters-alias the sample-covariance-based (SCB) minimum variance distortionless response beamformer output and, in general, the SCB linearly constrained minimum variance beamformer output-is likewise shown to be the same. These results have significant robustness implications for adaptive detection/estimation/beamforming in non-Gaussian environments.