On an analogue of Cramér-Rao's inequality
B. Raja Rao · Scandinavian Actuarial Journal · 1958
Rao [1] and simultaneously Cramér [2, 3] have shown that if f (x, θ) is the probability density function of a distribution involving an unknown parameter θ and distributed over the range α ⩽ x ⩽ b, where a and b are independent of θ, and if x 1 x 2 ... x n is a random sample of n independent observations from this distribution, the variance of any estimate unbiased for Ψ (θ), satisfies the inequality where E denotes mathematical expectation and is Fisher's information index about θ. In (1), equality holds if, and only if, θ* is sufficient for θ. This inequality is further generalized to the multi-parametric case.