generalized log-logistic distribution (e.g., El-Saidi, Singh, and Bartolucci, 1990).

This distribution is among the most commonly used models for the distribution of

personal incomes. In actuarial science it is usually called the Burr distribution (e.g.,

Hogg and Klugman, 1983, 1984).

The special case where a ¼ q is sometimes called the paralogistic distribution

(Klugman, Panjer, and Willmot, 1998). Further special cases are known as the Fisk

(1961) or log-logistic, for q ¼ 1, and the Lomax (1954) distribution, for a ¼ 1. They

will be discussed in greater detail in the following sections.

Unlike the c.d.f. of the GB2 distribution, the c.d.f. of the Singh “ Maddala

distribution is available in closed form; it is given by the pleasantly simple

expression

h xa iÀq

x . 0:

F(x) ¼ 1 À 1 þ , (6:43)

b

We directly determine that the quantile function is equally straightforward

F À1 (u) ¼ b[(1 À u)À1=q À 1]1=a , for 0 , u , 1: (6:44)

199

6.2 SINGH “ MADDALA DISTRIBUTION

Hence, the Singh “ Maddala distribution is one of the few distributions for which

the density, c.d.f., and quantile function all have simple closed forms. This may

partly explain why it was rediscovered so many times in various contexts.

Singh and Maddala (1976) derived their distribution by considering the hazard

rate r(x) of income. They observed that, although decreasing failure rate (DFR)

distributions are unlikely to be observed when the underlying variable is time, when

the variable is income, “ . . . a priori plausibility on theoretical reasoning for DFR

after a point is obvious” (p. 964), in that “income may help in earning more. The

ability to make more money might increase with one™s income.” [A similar idea may

be found some 50 years earlier in Hagstr“m (1925) in connection with the classical

Pareto distribution.]

Singh and Maddala then introduced the hazard rate of z :¼ log x, that is,

@F(z) 1

rÃ (z) ¼ ,

@z 1 À F(z)

a quantity they called the proportional failure rate (PFR). It measures, at any

income, the odds against advancing further to higher incomes, in a proportional

sense. For the classical Pareto distribution (3.2) we have rÃ (z) ¼ a. We know from

Chapter 3 that this is also the slope of the survival function of income in the Pareto

diagram. This suggests that rÃ (z) must be asymptotically constant. Singh and

Maddala further assumed that rÃ (z) grows with z ¬rst with an increasing, then a

decreasing, rate. De¬ning y :¼ Àlog(1 À F), y0 . 0, y00 . 0, they started from the

differential equation

y00 ¼ a Á y0 (a À y0 ),

where a is a constant. This may be rearranged, yielding

y00 y00

¼ aa,

þ

y0 a À y

which integrates to

log y0 À log(a À y0 ) ¼ aaz þ c1 ,

where c1 is the constant of integration. From this we get

y0

¼ eaazþc1 ,

0

aÀy

or

aeaazþc1

0

:

y¼

1 þ eaazþc1

200 BETA-TYPE SIZE DISTRIBUTIONS

Further integration yields

1

log y ¼ log(1 þ eaazþc1 ) þ c2 ,

a

where c2 is another constant of integration. Substituting Àlog(1 À F) for y, log x

for z, and rearranging, we obtain

c

F(x) ¼ 1 À ,

(b þ xaa )1=a

where c ¼ (Àc2 À c1 )=a and b ¼ 1=ec1 . The boundary condition F(0) ¼ 0 results in

c ¼ b1=a . Thus,

b1=a

F(x) ¼ 1 À ,

(b þ xaa )1=a

or

1

F(x) ¼ 1 À , (6:45)

(1 þ a1 xa2 )a3

where a1 ¼ 1=b, a2 ¼ aa, and a3 ¼ 1=a, which is the Singh “ Maddala c.d.f. given

above with a2 ¼ a, a3 ¼ q, and a1 ¼ ba2 .

This derivation shows that the Singh “Maddala distribution is characterized by

a proportional failure rate

aq Á eazÀlog b

Ã

r (z) ¼ ,

1 þ eazÀlog b

a three-parameter logistic function with respect to income power z ¼ log x. Singh

and Maddala™s approach was criticized by Cramer (1978) who found the analogy

with failure rates unconvincing, writing that “it is not clear what the DFR property of

income distributions means, but since it applies to distributions that hold at a given

time all references to the passage of time and to the process whereby individuals

move from one income level to the next are inappropriate.” Nonetheless, these

reliability properties seem to capture some salient features of empirical income

distributions.

The parameterization utilized in (6.45) is quite common in applications; a variant

with a1 ¼ 1=ba was employed, for example, by Schmittlein (1983). However, in the

˜

size distributions area the parameterization based on (6.42) is often more convenient,

because there b is a scale parameter and hence it is eliminated when scale-free

quantities such as the Lorenz curve and common inequality measures are

considered. We shall use this variant in the subsequent discussion.

201

6.2 SINGH “ MADDALA DISTRIBUTION

6.2.2 Moments and Other Basic Properties

The kth moment exists for Àa , k , aq; it equals [compare (6.12)]

bk B(1 þ k=a, q À k=a) bk G(1 þ k=a)G(q À k=a)

k

:

E(X ) ¼ (6:46)

¼

G(q)

B(1, q)

In particular,

bB(1 þ 1=a, q À 1=a) bG(1 þ 1=a)G(q À 1=a)

E(X ) ¼ (6:47)