From (5.12), the moments of the Weibull distribution are

k

E(X k ) ¼ bk G 1 þ , (5:81)

a

speci¬cally

1

E(X ) ¼ bG 1 þ , (5:82)

a

and

( !2 )

2 1

var(X ) ¼ b2 G 1 þ :

À G 1þ (5:83)

a a

The mode is at b(1 À 1=a)1=a if a . 1; otherwise, the distribution is zero-modal,

with a pole at the origin if a , 1.

The somewhat peculiar skewness properties of the Weibull distribution have been

studied by several authors; see for example, Cohen (1973), Rousu (1973), or

Groeneveld (1986). As is the case withp¬¬¬¬¬ generalized gamma distribution, there is

the

a value a for which the shape factor b1 ¼ 0. Unlike in the generalized gamma

case, this value does not depend on other parameters of the distribution; it equals (to

p¬¬¬¬¬

four decimals) a0 ¼ 3:6023. For a , a0 we have b1 . 0, while for a . a0

176 GAMMA-TYPE SIZE DISTRIBUTIONS

p¬¬¬¬¬

we have b1 , 0. Since empirical size distributions are heavily skewed to the right,

it would seem that a , a0 is the relevant range in our context.

The mean excess function is

°1

a a

e(x) ¼ e(x=b) eÀ(t=b) dt, x . 0, (5:84)

x

which is asymptotic to x1Àa (Beirlant and Teugels, 1992). [Incidentally, e(x) / x1Àa ,

0 , a 1, de¬nes the Benktander type II distribution, a loss distribution that will be

discussed in Section 7.4 below.]

The Weibull hazard rate

a x aÀ1

x . 0,

r(x) ¼ , (5:85)

bb

is a decreasing function when the shape parameter a is less than 1, a constant when

a equals 1 (the exponential distribution), and an increasing function when a is

greater than 1. The simple and ¬‚exible form of the hazard rate may explain why the

Weibull distribution is quite popular in lifetime studies.

A useful property of the Weibull order statistics is distributional closure of the

minima. Speci¬cally,

f1:n (x) ¼ n{(1 À F(x)}nÀ1 f (x)

& a '

na x aÀ1 x

x . 0:

, (5:86)

exp Àn

¼

bb b

Hence, X1:n $ Wei(a, bnÀ1=a ).

5.5.2 Lorenz Curve and Inequality Measurement

Lorenz-ordering relations are easily obtained using the star-shaped ordering (see

Section 2.1.1). Speci¬cally, we have for Xi $ Wei(ai , 1), i ¼ 1, 2, using (5.79)

F1 (u) {À log (1 À u)}1=a1

À1

, 0 , u , 1,

¼

F2 (u) {À log (1 À u)}1=a2

À1

which is seen to be increasing in u if and only if a1 a2 . Since the star-shaped

ordering implies the Lorenz ordering, we get (Chandra and Singpurwalla, 1981)

a2 :

X1 !L X2 ( a1 (5:87)

)

Hence, the Lorenz order is linear within the family of two-parameter Weibull

distributions. The result could, of course, also have been obtained directly from (5.19).

177

5.5 WEIBULL DISTRIBUTION

In view of (5.86), the Gini coef¬cient is most easily derived using the

representation in terms of order statistics (2.22), yielding

E(X1:2 )

¼ 1 À 2À1=a ,

G ¼1À (5:88)

E(X )

which is decreasing in a.

5.5.3 Estimation

Parameter estimation for the Weibull distribution is discussed in many sources,

notably in texts on engineering statistics. See also Cohen and Whitten (1988) and

Johnson, Kotz, and Balakrishnan (1994).

Brie¬‚y, the ML estimators satisfy the equations

2( 3À1

)( )À1

X X X

n n n

1

a¼4 log xi 5 ,

^ ^

xa log xi xa

^ (5:89)

À

i i

n i¼1

i¼1 i¼1

( )1=^

a

Xn

^¼ 1 ^

xa

b : (5:90)

i

n i¼1

^

Here (5.89) is solved for a; the result is then substituted in (5.90).

We note that only if a . 2, are we in a situation where the regularity conditions

for ML estimation are satis¬ed. In this case, the Fisher information on u ¼ (a, b)` is

given by

2 3

6(C À 1)2 þ p 2 CÀ1

6 b7

6a2

6 7

I (u) ¼ 6 7, (5:91)

4 25

CÀ1 a