)

Lomax (1954) considered it a suitable model for business failure data.

There is not as much variety in the possible basic shapes of the Pareto (II)

distribution”after all, it is just a shifted classical Pareto distribution”as with other

two-parameter models such as the gamma, Weibull, or Fisk, all of which allow for

zeromodal as well as unimodal densities. Its hazard rate is given by

q

x . 0,

r(x) ¼ , (6:132)

bþx

which is a strictly decreasing function for all admissible values of the parameters.

Nair and Hitha (1990) presented a characterization of the Lomax distribution

[under the name of the Pareto (II) distribution] in terms of the “equilibrium

distribution” de¬ned via the p.d.f. fZ (x) ¼ [1=E(X )]{1 À FX (x)} (this distribution is

of special signi¬cance in renewal theory). They showed that the condition eX (x) ¼

peZ (x) (where e denotes the mean excess function) for all x . 0, for some

0 , p , 1, characterizes the Lomax distribution. The distribution is also

characterized by the condition of proportionality of the corresponding hazard

rates, that is, by the condition hX (x) ¼ phZ (x) for all x . 0, for some p . 1.

The distribution can furthermore be characterized within the framework of a

model of underreported incomes (Revankar, Hartley, and Pagano, 1974); see

Section 3.4 for a detailed discussion. This result essentially exploits the linearity of

the mean excess function

xþb

x . 0:

e(x) ¼ , (6:133)

qÀ1

As in the case of the Fisk distribution, the Lorenz order is linear for Lomax

random variables, speci¬cally for Xi $ Lomax(bi , qi ), i ¼ 1, 2,

q1 q2 ( X1 !L X2 , (6:134)

)

228 BETA-TYPE SIZE DISTRIBUTIONS

provided qi . 1. The Gini coef¬cient is also of a very simple form

q

:

G¼ (6:135)

2q À 1

Note that 1=2 G 1, which casts some doubt on the usefulness of this model to

approximate income distributions of, for example, some European Union countries,

for which such extreme inequality is not observed.

Finally, we present some results pertaining to estimation.

P

For the scale parameter b ¼ 1, the statistic T ¼ n log(1 þ xi ) is suf¬cient and

i¼1

complete for qÀ1, and a minimum variance unbiased (MVU) estimator of q is given

by (n À 1)=T (Patel, 1973). The c.d.f. F(x) ¼ 1 À (1 þ x)Àq , x ! 0, can also be

estimated in an unbiased way; the estimator is

8 !nÀ1

>

< 1 À 1 À log(1 þ x) T . log(1 þ x)

,

^

F (x) ¼ ,

T

>

:

1, T log(1 þ x):

For further unbiased estimators of functions of Lomax parameters, see Voinov and

Nikulin (1993, pp. 435 “436), who refer to these results as results for the Burr”

meaning Burr XII, that is, Singh “ Maddala”distribution that involves an additional

shape parameter a. However, since all results given therein require a to be known, we

prefer to consider them as results pertaining to the Lomax subfamily. Algorithmic

aspects of ML estimation in the Lomax distribution (under the name of a Pareto

distribution) are discussed by Wingo (1979), who used a numerical method for the

univariate global optimization of functions expressible as the sum of a concave and a

convex function.

A generalization of the Lomax distribution was recently suggested by Zandonatti

(2001). Following Stoppa™s (1990b,c) approach leading to a generalized Pareto (I)

distribution (see Section 3.8), he arrived at a distribution with density

qu x ÀqÀ1 h x Àq iuÀ1

x . 0:

f (x) ¼ 1þ 1À 1þ , (6:136)

b b b

6.4.3 Empirical Applications

Incomes and Wealth

Fisk (1961a) considered weekly earnings in agriculture in England and Wales for

1955 “1956 and U.S. income distribution for 1954 (by occupational categories). He

concluded that the distribution may prove useful when income distributions that are

homogeneous in at least one characteristic (here occupation) are examined.

Using nonparametric bounds on the Gini coef¬cient developed by Gastwirth

(1972), Gastwirth and Smith (1972) found that the implied Gini indices derived from

229

6.4 FISK (LOG-LOGISTIC) AND LOMAX DISTRIBUTIONS

a Fisk distribution fall outside these bounds for U.S. individual adjusted gross

incomes for 1955 “ 1969 and concluded that Fisk distributions are inappropriate for

modeling these data.

Arnold and Laguna (1977) ¬t the Fisk distribution, with an extra location

parameter, to income data for 17 metropolitan areas in Peru for the period 1971 “

1972 and concluded that their results are reasonably consonant with a (shifted) Fisk

distribution. It is of historical interest to note that Vilfredo Pareto in his Cours d™

´

economie politique (1897) also used Peruvian income data (for the year 1800).

Harrison (1979) used the Fisk distribution (under the name of sech2 distribution;

see Chapter 7 for an explanation of this terminology) for the gross weekly earnings

of full-time male workers aged 21 and over, in Great Britain, collected in April 1972,

for seven occupational groups. The distribution performs about as well as a

lognormal distribution when the data are disaggregated, but considerably better in

the upper tail for the aggregate data.

McDonald (1984) ¬t the Fisk distribution to 1970, 1975, and 1980 U.S. family

incomes. However, the distribution does not do well: It is outperformed by the

(G)B1, (G)B2, Singh “ Maddala, and generalized gamma distributions, usually by

wide margins, and even the gamma and Weibull distributions (having the same

number of parameters) are preferable.

For the Japanese incomes (in grouped form) for 1963 “ 1971, the Fisk distribution

does only slightly worse than the Singh “ Maddala and outperforms the beta, gamma,

lognormal, and Pareto (II) distributions (Suruga, 1982). In fact, a likelihood ratio test

reveals that there are no signi¬cant differences between the Fisk and Singh “

Maddala distributions for these data; hence, the simpler Fisk distribution is entirely

adequate. The distribution was also ¬tted to several strata from the 1975 Japanese

Income Redistribution Survey by Atoda, Suruga, and Tachibanaki (1988). Four

occupational classes as well as primary and redistributed incomes were considered,

and ¬ve different estimation techniques applied. In a later study using the same data

set, Tachibanaki, Suruga, and Atoda (1997) applied ML estimation on the basis of

individual observations. In both studies, it became clear that the data require a more

¬‚exible model such as the Singh “ Maddala distribution. Among the two-parameter

functions, gamma and Weibull distributions both ¬t better.

Henniger and Schmitz (1989) considered the Fisk distribution when com-

paring ¬ve parametric models for the UK Family Expenditure Survey for 1968 “

1983 to nonparametric ¬ttings. Although for the entire population all parametric

models are rejected, the Fisk distribution performs reasonably well for some

subgroups.

Actuarial Losses

In the actuarial literature, Benckert and Jung (1974) employed the Pareto (II)

distribution to model the distribution of ¬re insurance claims in Sweden for the

period 1958 “1969. They found that for one class of buildings (wooden houses) the

distribution provides a good ¬t, with estimates of the tail index q less than 1 in all

cases.