Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICSand describes the spread of values â about E[â] that would result from a largenumber of samples, each of size N. An estimator with a smaller variance is saidto be more efficient than one with a larger variance. As we show in the nextsection, for any given quantity a of the population there exists a theoretical lowerlimit on the variance of any estimator â. This result is known as Fisher’s inequality(or the Cramér–Rao inequality) and readsV [â] ≥(1+ ∂b∂a) 2 /E[ ]− ∂2 ln P∂a 2 , (31.17)where P stands for the population P (x|a) andb is the bias of the estimator.Denoting the quantity on the RHS of (31.17) by V min ,theefficiency e of anestimator is defined ase = V min /V [â].An estimator for which e = 1 is called a minimum-variance or efficient estimator.Otherwise, if e