G(q)

B(1, q)

and

b2 {G(q)G(1 þ 2=a)G(q À 2=a) À G2 (1 þ 1=a)G2 (q À 1=a)}

:

var(X ) ¼ (6:48)

G2 (q)

Hence, the coef¬cient of variation is (McDonald, 1981)

s¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

G(q)G(1 þ 2=a)G(q À 2=a)

CV ¼ À1 (6:49)

G2 (1 þ 1=a)G2 (q À 1=a)

and the shape factors are

p¬¬¬¬¬ G2 (q)l3 À 3G(q)l2 l1 þ 2l3

1

b1 ¼ (6:50)

2 3=2

[G(q)l2 À l1 ]

and

G3 (q)l4 À 4G2 (q)l3 l1 þ 6G(q)l2 l2 À 3l4

1 1

b2 ¼ , (6:51)

22

[G(q)l2 À l1 ]

where we have set li ¼ G(1 þ i=a)G(q À i=a), i ¼ 1, 2, 3, 4. These expressions are

rather unwieldy.

From (6.11), the mode of the Singh “Maddala distribution is at

1=a

aÀ1

if a . 1,

xmode ¼b , (6:52)

aq þ 1

and at zero otherwise. Thus, the mode is seen to be decreasing with q, re¬‚ecting the

fact that the right tail becomes lighter as q increases.

202 BETA-TYPE SIZE DISTRIBUTIONS

The closed forms of the c.d.f and the quantile function of the Singh “ Maddala

distribution also permit convenient manipulations with the characteristics of order

statistics. In particular,

bG(1 þ 1=a)G(nq À 1=a)

:

E(X1:n ) ¼ (6:53)

G(nq)

This follows from the closure property (Arnold and Laguna, 1977; Arnold, 1983)

X $ SM(a, b, q) ¼ X1:n $ SM(a, b, nq) (6:54)

)

and the expression for the moments (6.46).

The moments of Xk:n , 2 k n can now be generated using recurrence relations,

yielding (Tadikamalla, 1977)

n kX

iÀ1

k k

iÀ1

(À1) j

k

B qn À qi þ q À þ qj, 1 þ :

E(Xi:n ) ¼ i bq

i j a a

j¼0

However, expectation of minima is all we need for the derivation of the generalized

Gini coef¬cients below.

Arnold and Laguna (1977) provided tables of E(Xk:n ), 1 k n, for n ¼ 1(1)10

and parameter values of b ¼ 1, q ¼ 0:5(0:5)5:0, and aÀ1 ¼ 0:1(0:1)1:0.

Since the c.d.f. is available in a simple closed form, this is also the case for the

hazard rate, which is given by

aqxaÀ1

, x . 0:

r(x) ¼ (6:55)

ba {1 þ (x=b)a }

The general shape of this function depends on the value of the shape parameter a:

For all a . 0 the hazard rate is eventually decreasing. For a . 1 we have a unimodal

function, whereas for a 1 it is decreasing for all x . 0. [The special case where

a ¼ 2 (which is associated with a unimodal hazard rate) is discussed by Greenwich

(1992) in some detail.] Note that the parameter q is only a scale factor in (6.55) and

does not determine the shape of the function.

From general results for the mean excess function of regularly varying

distributions (see Chapter 2), we determine that the mean excess function is

asymptotically linearly increasing

e(x) [ RV1 (1): (6:56)

6.2.3 Representations and Characterizations

Several characterizations of the Singh “ Maddala distribution are available in the

statistical and econometric literature. Most of them are probably best understood by

relating them to characterizations of the exponential distribution, the most widely

203

6.2 SINGH “ MADDALA DISTRIBUTION

known and used distribution on IRþ . The exponential distribution is remarkably well

behaved, in that its c.d.f., quantile function, m.g.f., mean excess function, etc. are

all available in simple closed form. Not surprisingly, it has generated a substantial

characterizations literature. For a more extensive discussion of exponential

characterizations, see Galambos and Kotz (1978) for results until the late 1970s

and Johnson, Kotz, and Balakrishnan (1994, Section 19.8) for selected results that

were obtained thereafter.

For any monotonic function h(Á), the characterization of X is equivalent to that of

h(X ). Thus, if a speci¬c distribution can be associated with the exponential

distribution, a host of characterization results become available. The Singh “

Maddala distribution is related to the exponential distribution via

a !

X

X $ SM(a, b, q) ( log 1 þ $ Exp(q), (6:57)

)

b

where Exp(q) denotes an exponential distribution with scale parameter q. Hence,

h x a i

, a . 0,

h(x) ¼ log 1 þ

b

is the required monotonic transformation, and many characterizations of Singh “

Maddala distributions are now available by applying the transformation hÀ1 (Á) on

exponential variables.

The lack of memory property is perhaps the most popular and intuitively

transparent characterization of the exponential distribution. One of its many

equivalent expressions is given in terms of the functional equation

P(X . x þ y) ¼ P(X . x)P(X . y): (6:58)

El-Saidi, Singh, and Bartolucci (1990) showed that the functional equation

a ! a ! a !

X X X

. xy ¼ P 1 þ .x ÁP 1þ .y ,

P 1þ

b b b

where a, b . 0 and x, y . 1, characterizes the Singh “Maddala distribution (which

they call a generalized log-logistic distribution) among all continuous distributions

supported on [0, 1). Ghitany (1996) pointed out that this result, and others to be

discussed below, derive essentially from the lack of memory property (6.58) of the

exponential distribution. For the remainder of this section we shall ignore the scale