Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 285produces a moment estimator ^ for in the form^ ˆ 2 1=2: …9:82†M 2Although this estimator may be inferior to 1/X in terms of the quality criteriawe have established, an interesting question arises: given two or more momentestimators, can they be combined to yield an estimator superior to any of theindividual moment estimators?In what follows, we consider a combined moment estimator derived from anoptimal linear combination of a set of moment estimators. Let ^(1) , ^ (2) ,...,^ p) be p moment estimators for the same parameter . We seek a combinedestimator in the form p p ˆ w 1 ^ …1† ‡‡w p ^…p† ;…9:83†where coefficients w 1 ,..., and w p are to be chosen in such a way that it isunbiased if ^ j) , j ˆ 1, 2, . . . , p, are unbiased and the variance of p is minimized.The unbiasedness condition requires thatw 1 ‡‡w p ˆ 1:…9:84†We thus wish to determine coefficients w j by minimizingQ ˆ varf p gˆvar )subject to Equation (9.84).Let u T ˆ [1 1], ^QT ˆ [ ^ 1) ^p) ], and w T ˆ [w 1 w p ].Equations (9.84) and (9.85) can be written in the vector–matrix formX pjˆ1w j ^…j†; …9:85†andQ…w† ˆvarw T u ˆ 1; )X pjˆ1w j ^…j†ˆ w T w;…9:86†…9:87†where ˆ [ ij ]with ij ˆ covf ^ i) , ^ j) g.In order to minimize Equation (9.87) subject to Equation (9.86), we considerQ 1 …w† ˆw T w w T u u T w …9:88†TLFeBOOK