Introduction to Probability and Statistics for Engineers and Scientists

602 Chapter 14*:Life Testing

or, equivalently,

αˆ=

n

∑n

i= 1

x

βˆ

i

n+βˆlog

(n

∏

i= 1

xi

)

=

nβˆ

∑n

i= 1

x

βˆ

i logxi

∑n

i= 1

x

βˆ

i

This latter equation can then be solved numerically forβˆ, which will then also determineαˆ.
However, ratherthanpursuingthisapproachanyfurther, letusconsiderasecondapproach,
which is not only computationally easier but appears, as indicated by a simulation study,
to yield more accurate estimates.

14.5.1 Parameter Estimation by Least Squares

LetX 1 ,...,Xnbe a sample from the distribution

F(x)= 1 −e−αx

β

, x≥ 0

Note that

log(1−F(x))=−αxβ

or

log

(

1

1 −F(x)

)

=αxβ

and so

log log

(

1

1 −F(x)

)

=βlogx+logα (14.5.3)

Now letX(1)<X(2)< ···<X(n)denote the ordered sample values — that is, for
i=1,....n,

X(i)=ith smallest ofX 1 ,...,Xn

and suppose that the data results inX(i)=x(i). If we were able to approximate the quantities
log log(1/[ 1 −F(x(i))]) — say, by the valuesy 1 ,...,yn— then from Equation 14.5.3, we