Introduction to Probability and Statistics for Engineers and Scientists

14.5The Weibull Distribution in Life Testing 603

could conclude that

yi≈βlogx(i)+logα, i=1,...,n (14.5.4)

We could then chooseαandβto minimize the sum of the squared errors — that is,α
andβare chosen to

minimize

α,β

∑n

i= 1

(yi−βlogx(i)−logα)^2

Indeed, using Proposition 9.2.1 we obtain that the preceding minimum is attained when
α=ˆα,β=βˆwhere

βˆ=

∑n

i= 1

yilogx(i)−nlogxy ̄

∑n

i= 1

(logx(i))^2 −n(logx)^2

logαˆ= ̄y−βlogx

where

logx=

∑n

i= 1

(logx(i))

/

n, y ̄=

∑n

i= 1

yi

/

n

To utilize the foregoing, we need to be able to determine valuesyithat approximate log
log(1/[ 1 −F(x(i))]=log[−log(1−F(x(i)))),i=1,...,n. We now present two different
methods for doing this.
Method 1:This method uses the fact that

E[F(X(i))]=

i

(n+1)

(14.5.5)

and then approximatesF(x(i))byE[F(X(i))]. Thus, this method calls for using

yi=log{−log(1−E[F(X(i))])} (14.5.6)

=log

{

−log

(

1 −

i

(n+1)

)}

=log

{

−log

(

n+ 1 −i

n+ 1

)}