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

as a condition of extreme. SinceδwandδwTare arbitrary, we require that

and either of these two relations gives

The constraint Equation (9.86) is now used to determine. It implies that

or

H ence, we have from Equations (9.90) and (9.91)

The variance of is

in view of Equation (9.92).
Several attractive features are possessed by For example, we can show
that its variance is smaller than or equal to that of any of the simple moment
estimators 1,2,...,p, and furthermore (see Soong, 1969),

if p q.

Example 9.14.Consider the problem of estimating parameter in the log-
normal distribution

from a sample of size n.

286 Fundamentals of Probability and Statistics for Engineers



dQ 1 …w†ˆdwT…wu†‡…wTuT†dwˆ 0

wuˆ 0 and wTuTˆ 0 ; … 9 : 89 †

wTˆuT^1 : … 9 : 90 †



wTuˆuT^1 uˆ 1 ;

ˆ

1

uT^1 u

: … 9 : 91 †

wTˆ

uT^1

uT^1 u

: … 9 : 92 †

^ p

varf^ pgˆwTwˆ

1

uT^1 u

; … 9 : 93 †

^ p.

^j),jˆ

varf^ pgvarf^ qg; … 9 : 94 †





f…x;†ˆ

1

x… 2 †^1 =^2

exp

1

2 

ln^2 x



; x 0 ;> 0 ; … 9 : 95 †