Further Singh “ Maddala characterizations are in terms of order statistics. Suppose

that X1 , . . . , Xn are a random sample from an absolutely continuous distribution with

c.d.f. F and let X1:n , . . . , Xn:n denote their order statistics. Al-Hussaini (1991)

showed that Xi $ SM(a, 1, q), i ¼ 1, . . . , n, if and only if, for some a . 0,

a a a

1 þ X2:n 1 þ X3:n 1 þ Xn:n

a

a , . . . , Z2 ¼

Z1 ¼ 1 þ X1:n , Z2 ¼ a , Z2 ¼ a

1 þ X1:n 1 þ X2:n 1 þ XnÀ1:n

204 BETA-TYPE SIZE DISTRIBUTIONS

are independent. Since the exponential distribution is characterized by the

independence of successive spacings, this result can also be traced back to the

exponential case via the transformation h(Á) de¬ned above [Ghitany (1996), who was

however unaware of Al-Hussaini™s (1991) result].

Another group of characterizations is related to the mean excess (or mean residual

life) function. Dimaki and Xekalaki (1996) showed that the Singh “ Maddala

distribution is characterized by the property

E[log(1 þ X a )jX . y] ¼ log(1 þ ya ) þ c, (6:59)

for all nonnegative values of y and for some a . 0, c . 0 among the continuous

distributions with the support on [0, 1) and such that Ejh(X )j , 1. This is

essentially a restatement of the condition E(X jX . y) ¼ y þ E(X ), for all non-

negative y, which is known to characterize the exponential distribution (Shanbhag,

1970).

Fakhry (1996) provided three related results. First, for h(x) ¼ log(1 þ xa ) the

recurrence relation

1

E[hk (X )jX . y] ¼ hk ( y) þ kE[hkÀ1 (X )jX . y] (6:60)

q

characterizes the SM(a, 1, q) distribution. Under the additional condition that

Eh2 (X ) , 1, the same is true if, for all y and some ¬xed c,

var[h(X )jX . y] ¼ c2 : (6:61)

Third, he observed that the condition

k

E[hk (Xi:n )] ¼ E[hk (XiÀ1:n )] þ E[hkÀ1 (Xi:n )] (6:62)

q(n À i þ 1)

also characterizes the SM(a, 1, q) distribution. Again, all three results are connected

to characterizations of the exponential distribution via the transformation h.

Characterization (6.60) is a higher-order moment version of (6.59). The exponential

characterization associated with (6.61) is due to Azlarov, Dzamirzaev, and Sultanova

(1972) and Laurent (1974), whereas the characterization (6.62) via relationships

between the moments of order statistics is reduced to an exponential characterization

provided by Lin (1988).

Khan and Khan (1987) presented further characterizations based on conditional

expectations of order statistics. If a ¼ q(n À i) is independent of x,

!

a 1

k

xk þ

E[Xiþ1:n jXi:n ¼ x] ¼ (6:63)

aÀ1 a

205

6.2 SINGH “ MADDALA DISTRIBUTION

characterizes the Singh “ Maddala distribution (referred to as the Burr XII by Khan

and Khan) among all continuous distributions with support on [0, 1) and F(0) ¼ 0.

In view of E[Xn:n jXnÀ1:n ¼ x] ¼ E[X k jX . x] and E[X1:n jX2:n ¼ y] ¼ E[X k jX y],

k k

this can be expressed alternatively as

!

q 1

E[X k jX ! x] ¼ xk þ :

qÀ1 q

Also, the distribution is characterized by

1 q 1 À F(x) k

k

x:

E[X1:n jX2:n ¼ x] ¼ (6:64)

À

q À 1 q À 1 F(x)

Both results can be restated in terms of E[X k jX1:n ¼ x] or E[X k jXn:n ¼ x].

Specializing from the mixture representation of the GB2 distribution (6.21), the

Singh “Maddala distribution can be considered a compound Weibull distribution

whose scale parameter follows an inverse generalized gamma distribution (Takahasi,

1965; Dubey, 1968)

^

Wei(a, u) InvGG(a, b, q) ¼ SM(a, b, q): (6:65)

u

6.2.4 Lorenz Curve and Inequality Measures

As we have already mentioned, the quantile function of the Singh “ Maddala

distribution is available in closed form. Consequently, its (normalized) integral, the

Lorenz curve, is also of a comparatively simple form, namely,

1 1

L(u) ¼ Iy 1 þ , q À , 0 u 1, (6:66)

a a

where y ¼ 1 À (1 À u)1=q . The subclass where q ¼ (a þ 1)=a is even simpler

analytically and yields

L(u) ¼ [1 À (1 À u)a=(aþ1) ](aþ1)=a , 0 , u , 1, (6:67)

a Lorenz curve that is symmetric (in the sense of Section 2.1.1). It is interesting that

this is also a subclass of the Rasche et al. (1980) family of Lorenz curves; cf. Section

2.1. Thus, the Rasche et al. model and the Singh “ Maddala distribution share the

same underlying structure.

For Xi $ SM(ai , bi , qi ), i ¼ 1, 2, the necessary and suf¬cient conditions for

X1 !L X2 are

a2 q2 :

a1 a2 and a1 q1 (6:68)

[See (6.24) and (6.25).] This result is due to Wil¬‚ing and Kramer (1993).

¨

206 BETA-TYPE SIZE DISTRIBUTIONS

From (6.53) we obtain the generalized Gini coef¬cients (Kleiber and Kotz, 2002)

G(nq À 1=a)G(q)

E(X1:n )

Gn ¼ 1 À ¼1À

G(nq)G(q À 1=a)

E(X )

for n ¼ 2, 3, 4, . . . , where n ¼ 2 yields the ordinary Gini coef¬cient (Cronin, 1979;

McDonald and Ransom, 1979a)

G(q)G(2q À 1=a)

:

G ¼1À (6:69)

G(q À 1=a)G(2q)

The Theil measure is (McDonald, 1981)

!

1 1 1 1 1

À c ° q þ 1Þ À c q À

T (X ) ¼ 2c À log(q) À log B 1 þ , q À ,

a a a a a

and the Pietra index can be written, using (6.23) and the representation P ¼ F(m)À

F(1) (m) (Butler and McDonald, 1989),