Limitations on SNR estimator accuracy

We consider the samples of a pure tone in additive white Gaussian noise (AWGN) for which we wish to determine the signal-to-noise ratio (SNR) defined here to be α=(A²/2σ²), where A is the tone amplitude, and σ² is the noise variance. A and σ² are assumed to be deterministic but unknown a priori. If the variance of an unbiased estimator of α is σ_αˆ², we show that at high SNR, the normalized standard a deviation satisfies the Cramer-Rao lower bound (CRLB) according to σ_αˆ/α≥√(2/N), where N is the number of independent observables used to obtain the SNR estimate σˆ.