Fundamentals of Probability and Statistics for Engineers

produces a moment estimator for in the form

Although this estimator may be inferior to 1/X in terms of the quality criteria
we have established, an interesting question arises: given two or more moment
estimators, can they be combined to yield an estimator superior to any of the
individual moment estimators?
In what follows, we consider a combined moment estimator derived from an
optimal linear combination of a set of moment estimators. Let (1), (2),...,
be p moment estimators for the same parameter. We seek a combined
estimator in the form

where coefficients w 1 ,..., and wp aretobechoseninsuchawaythatitis
unbiased if 1, 2,... , p, are unbiased and the variance of is minimized.
The unbiasedness condition requires that

We thus wish to determine coefficients wj by minimizing

subject to Equation (9.84).

Equations (9.84) and (9.85) can be written in the vector–matrix form

and

where
In order to minimize Equation (9.87) subject to Equation (9.86), we consider

Parameter Estimation 285

^ 

^ˆ^2

M 2

 1 = 2

: … 9 : 82 †

^ ^

^p) 
^ p

^ pˆw 1 ^…^1 †‡‡wp^…p†; … 9 : 83 †

^j),jˆ ^

p

w 1 ‡‡wpˆ 1 : … 9 : 84 †

Qˆvarf^ pgˆvar

Xp

jˆ 1

wj^…j†

)

; … 9 : 85 †

Let uTˆ[1  1],Q^Tˆ[^1)  ^p)], and wTˆ[w 1  wp].

wTuˆ 1 ; … 9 : 86 †

Q…w†ˆvar

Xp

jˆ 1

wj^…j†

)

ˆwTw; … 9 : 87 †

ˆ[ij]withijˆcovf^i),^j)g.

Q 1 …w†ˆwTwwTuuTw … 9 : 88 †