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Signal convexity and noise convexity of the Chernoff and divergence distances

A number of convexity properties for the Chernoff and divergence distances between the statistical hypotheses--Gaussian noise versus Gaussian noise plus additive Gaussian signal--are considered. It is shown that 1) both the divergence and Chernoff distances are convex with respect to the noise covariance; 2) the divergence is convex with respect to the signal covariance; and 3) the Chernoff distance is convex with respect to the signal covariance provided that a low signal-to-noise variance ratio criterion is satisfied.