Page 303 - Fundamentals of Probability and Statistics for Engineers

286 Fundamentals of Probability and Statistics for Engineers

where is the Lagrange multiplier. Taking the first variation of Equation (9.88)
and setting it to zero we obtain

T
T
T
dQ w†ˆ dw w u †‡ w u †dw ˆ 0
1
T
as a condition of extreme. Since δw and δw are arbitrary, we require that
T
T
w u ˆ 0 and w u ˆ 0; 9:89†
and either of these two relations gives
T
T
1
w ˆ u : 9:90†

The constraint Equation (9.86) is now used to determine . It implies that
1
T
T
w u ˆ u u ˆ 1;
or
1
ˆ 1 : 9:91†
u u
T
Hence, we have from Equations (9.90) and (9.91)
T
u 1
T
w ˆ 1 : 9:92†
u u
T
The variance of p is
1
T

varf gˆ w w ˆ ; 9:93†
p T 1
u u
in view of Equation (9.92).

Several attractive features are possessed by . For example, we can show
p
that its variance is smaller than or equal to that of any of the simple moment
^ j)
estimators , j ˆ 1, 2, .. . , p, and furthermore (see Soong, 1969),

varf g varf g; 9:94†
p q
if p q.
Example 9.14. Consider the problem of estimating parameter in the log-
normal distribution
1 1 2
f x; †ˆ exp ln x ; x 0; > 0; 9:95†
x 2 † 1=2 2
from a sample of size n.

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