Page 302 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 285
^
produces a moment estimator for in the form

2 1=2
^
ˆ : 9:82†
M 2
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) ,...,
^ p)
be p moment estimators for the same parameter . We seek a combined

estimator p in the form
^ p†
^ 1†

ˆ w 1 ‡ ‡ w p ; 9:83†
p
where coefficients w 1 , .. ., and w p are to be chosen in such a way that it is
^ j)
unbiased if , j ˆ 1, 2, . . . , p, are unbiased and the variance of is minimized.
p
The unbiasedness condition requires that
w 1 ‡ ‡ w p ˆ 1: 9:84†
We thus wish to determine coefficients w j by minimizing

p
)

^ j†
X
Q ˆ varf gˆ var w j ; 9:85†
p
jˆ1
subject to Equation (9.84).
^ p)
^ T
^ 1)
T
T
Let u ˆ [1 1], Q ˆ [ ], and w ˆ [w 1 w p ].
Equations (9.84) and (9.85) can be written in the vector–matrix form
T
w u ˆ 1; 9:86†
and
)
p
X T
^ j†
Q w†ˆ var w j ˆ w w; 9:87†
jˆ1
^ i) ^ j)
where ˆ [ ij ]with ij ˆ covf , g.
In order to minimize Equation (9.87) subject to Equation (9.86), we consider
T
T
T
Q w†ˆ w w w u u w 9:88†
1

TLFeBOOK