On Cramer-Rao bounds for mean-square errors of estimators with linear expectations

Let x 1, ..., x n be a random sample from a density σ -1 f((x-μ)/gs, where f is known but μ ∈ iR and σ>0 unknown. A familiar multiparameter version of the Cramér-Rao theorem asserts that under regularity conditions on f of standard type the inverse of the Fisher information matrix I is a lower bound for the covariance matrix V of unbiased estimators μ* and σ* of μ and σ (see Cramér, 1946 and Rao, 1973, pp. 326–328). If for particular unbiased estimators μ* and σ* we find that V is close to I -1, then we know that μ* and σ* are good unbiased estimators and the C-R inequality is very informative.