households in receipt of social bene¬ts using the 1979 UK Family Expenditure

Survey (FES) and to incomes from the 1985 UK FES. She employed the MLE as

well as an optimal B-robust estimator (OBRE) and concluded that the latter provides

222 BETA-TYPE SIZE DISTRIBUTIONS

a better ¬t for the bulk of the 1979 data. The Dagum distribution is also preferred

over the gamma distribution here. For the 1985 data, however, the differences

between the two estimators are insigni¬cant and the Dagum distribution does not do

appreciably improve on the gamma.

Bantilan et al. (1995) modeled incomes from the Family Income and Expenditure

Surveys (FIES) in the Philippines for 1957, 1961, 1965, 1971, 1985, and 1988 using

a Dagum type I distribution. They noted that the model ¬ts the data rather well,

particularly in the tails.

Bordley, McDonald, and Mantrala (1996) ¬t the Dagum type I distribution to U.S.

family incomes for 1970, 1975, 1980, 1985, and 1990. For all data sets it turns out to be

the best three-parameter model, being inferior only to the GB2 distribution and an

observationally equivalent generalization and outperforming three- and four-parameter

models such as the generalized gamma and GB1 distributions by wide margins.

Botargues and Petrecolla (1997, 1999a,b) estimated Dagum type I “III models for

income distribution in the Buenos Aires region, for all years from 1990“ 1997. They

found that the Dagum models outperform lognormal and Singh “ Maddala

distributions, sometimes by wide margins.

Actuarial Losses

In the actuarial literature, Cummins et al. (1990) ¬t the Dagum type I distribution

(under the name of Burr III) to aggregate ¬re losses from Cummins and Freifelder

(1978). The distribution performs rather well; however, its full ¬‚exibility is not

required and a one-parameter limiting case, the inverse exponential, is fully adequate.

The same paper also considered data on the severity of ¬re losses and ¬t the GB2 and

its subfamilies to both grouped and individual observations. Although the ¬t of the

Dagum model is very good, that of the Singh “ Maddala distribution is slightly better.

6.4 FISK (LOG-LOGISTIC) AND LOMAX DISTRIBUTIONS

In this section we collect some results on the one- and two-parameter subfamilies of

the GB2 distribution. Although these models may not be suf¬ciently ¬‚exible in the

present context, they have also been considered in various applications and some

results pertaining to them”but not to their generalizations”are available. Since the

moments and other basic properties have been presented in a more general form in

the previous sections, we shall be brief and mention only those results that have not

been extended to more general distributions.

6.4.1 Fisk Distribution

The ¬rst of these distributions is the Fisk distribution with c.d.f.

h x a iÀ1 h x Àa iÀ1

x . 0,

F(x) ¼ 1 À 1 þ ¼ 1þ , (6:121)

b b

223

6.4 FISK (LOG-LOGISTIC) AND LOMAX DISTRIBUTIONS

and p.d.f.

axaÀ1

x . 0,

f (x) ¼ a , (6:122)

b [1 þ (x=b)a ]2

where a, b . 0, a Singh “Maddala distribution with q ¼ 1. Alternatively, it can be

considered a Dagum distribution with p ¼ 1. This model is also a special case of the

three-parameter Champernowne distribution to be discussed in Chapter 7 and was

actually brie¬‚y considered by Champernowne (1952). However, in view of the more

extensive treatment by Fisk (1961a,b), it is usually called the Fisk distribution in

the income distribution literature. Some authors, for example, Dagum (1975) and

Shoukri, Mian, and Tracy (1988), refer to the Fisk distribution as the log-logistic

distribution, whereas Arnold (1983) calls it a Pareto (III) distribution and includes an

additional location parameter. The term log-logistic may be explained by noting that

the distribution of log X is logistic with scale parameter a and location parameter

log b.

A useful property of the Fisk distribution is that it allows for nonmonotonic

hazard rates, speci¬cally

axaÀ1

x . 0,

h(x) ¼ a , (6:123)

b [1 þ (x=b)a ]

which is decreasing for a 1 and unimodal with the mode at x ¼ b(a À 1)1=a

otherwise. Among the distributions discussed in the present chapter, the Fisk

distribution is the simplest model with this property. In contrast, more popular

two-parameter distributions such as the Weibull only allow for monotonic hazard

rates.

Dagum (1975) considered a mixture of this distribution with a point mass at the

origin, a model that may be viewed as a predecessor of the Dagum type II

distribution considered in the previous section.

Interestingly, the distributions of the order statistics from a Fisk distribution have

been encountered earlier in this chapter: For a Fisk(a,b) parent distribution, the p.d.f.

of Xi:n is

n

F(x)iÀ1 [1 À F(x)]nÀi f (x)

fi:n (x) ¼ i

i

n!axaiÀ1

: (6:124)

¼

(i À 1)!(n À i)!bai [1 þ (x=b)a ]nþ1

This can be recognized as the p.d.f. of a GB2 distribution, speci¬cally, Xi:n $

GB2(a, b, i, n À i þ 1) (Arnold, 1983, p. 60). [Much earlier, Shah and Dave (1963)

already presented percentile points of log-logistic order statistics for n ¼ 1(1)10,

1 k n, when a ¼ b ¼ 1.]

224 BETA-TYPE SIZE DISTRIBUTIONS

Since b is a scale and a is the only shape parameter, we cannot expect much

¬‚exibility in connection with inequality measurement. For the Fisk distributions the

Lorenz order is linear, speci¬cally for Xi $ Fisk(ai , bi ), i ¼ 1, 2, we get from (6.24)

X1 !L X2 ( a1 a2 , (6:125)

)

provided that ai . 1, i ¼ 1, 2. It should also be noted that the expression for the Gini

coef¬cient is even simpler than for the classical Pareto distribution; it is just

1

G¼ : (6:126)

a

The Fisk distribution is characterized by

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, among all continuous distributions supported on [0, 1)

(Shoukri, Mian, and Tracy, 1988). As discussed above in Sections 6.2 and 6.3, this

was generalized to the Singh “Maddala and Dagum distributions by El-Saidi, Singh,

and Bartolucci (1990) and is intimately related to the lack of memory property of the

exponential distribution.

Following the earlier work of Arnold and Laguna (1977), Arnold, Robertson, and

Yeh (1986) provided a characterization of the Fisk distribution (which they call a

Pareto (III) distribution) in terms of geometric minimization. Suppose Np is a

geometric random variable independent of Xi $ Fisk(a, b), i ¼ 1, 2, . . . , with

P(Np ¼ i) ¼ p(1 À p)iÀ1 , i ¼ 1, 2, . . . , for some p [ (0, 1). Then

1

1=a

Up ¼ min Xi $ Fisk bp ,

a

i Np