1

Vp ¼ max Xi $ Fisk bpÀ1=a , ,

a

i Np

that is,

pÀ1=a Up $ p1=a Vp $ X1 : (6:127)

Under the regularity condition limx#0þ xÀa F(x) ¼ bÀa , it follows that if any one

of the statements in (6.127) holds for a ¬xed p [ (0, 1) then Xi $ Fisk(a, b). [This

mechanism has been generalized in several directions by Pakes (1983), among other

things relaxing the assumption that Np be geometrically distributed. However, in

225

6.4 FISK (LOG-LOGISTIC) AND LOMAX DISTRIBUTIONS

general it is no longer possible to obtain the resulting income distribution in

closed form.]

Regarding estimation, it is worth noting that the Fisher information of the Fisk

model is diagonal; thus, the distribution is characterized by orthogonal parameters.

This is a direct consequence of the fact that the distribution of Y ¼ log X is a

location-scale family, afY [( y À log b)a], with an fY that is symmetric about the origin

(clearly, fY is logistic). See Lehmann and Casella (1998), Example 2.6.5. For Fisk

distributions there is therefore no loss of asymptotic ef¬ciency in estimating a or b

when the other parameter is unknown. This property does not extend to the three-

and four-parameter distributions discussed in this chapter.

Shoukri, Mian, and Tracy (1988) considered probability-weighted moments

estimation of the Fisk model parameters. The probability-weighted moments

(PWMs) are de¬ned as

Wl, j,k ¼ E{X l {F(X )}j [1 À F(X )]k },

where l, j, k are real numbers. For l ¼ 1 and k ¼ 0, Wr ¼ W1,r,0 ¼ E[XF(X )r ] will

denote the PWMs of order r. For the Fisk distribution,

bG(r þ 1 þ 1=a)G(1 À 1=a)

:

Wr ¼

G(r þ 2)

In particular,

bp pÀ1

1 1

W0 ¼ bG 1 þ G 1 À (6:128)

sin

¼

a a a a

and

1þa

W0 :

W1 ¼ (6:129)

a

Given a complete random sample of size n, the estimation of Wr is most

conveniently based on the order statistics. The statistic

1X YjÀi

n r

^

Wr ¼ xj:n

n j¼1 nÀi

i¼1

is an unbiased estimator of Wr (Landwehr, Matalas, and Wallis, 1979). The PWM

estimators are now solutions of (6.128) and (6.129) when the Wr are replaced by

^

their estimators Wr . Thus,

^

W0

aÃ ¼

^ ^

2W 1 À W 0

226 BETA-TYPE SIZE DISTRIBUTIONS

and

^2

W0 sin(p=aÃ )

bÃ ¼

^ ^

p(2W1 À W0 )

are the PWM estimators of the parameters a and b, respectively. Shoukri, Mian, and

Tracy presented the asymptotic covariance matrix of the estimators that can be

derived from the general properties of statistics representable as linear functions of

order statistics. They also showed that for the parameter a the PWM estimator is

asymptotically less biased than the ML estimator for a ! 4, and that for a . 7 each

parameter estimator has asymptotic ef¬ciency of more than 90% relative to the

MLE. A small simulation study comparing PWM and ML estimators for samples of

size n ¼ 15 and n ¼ 25 shows that the PWMs compare favorably with the MLEs:

The PWN estimators seem to be less biased and almost consistently have smaller

variances. In addition, for the Fisk distribution the PWM estimators are fast and

straightforward to compute and always yield feasible values for the estimated

parameters. However, for shape parameters a 6 the MLE is generally more

ef¬cient, and this seems to be the relevant range in the present context.

Chen (1997) derived exact con¬dence intervals and tests for the Fisk shape

parameter a. Observing that the distribution of the ratio j ¼ MA =MG of the

arithmetic mean and the geometric mean of

X1:n a X2:n a Xk:n a

,...,

,

b b b

is parameter-free with a strictly increasing c.d.f., he obtained percentile points for

3 k n 30 by a Monte Carlo simulation. These tables can also be used to

perform tests on the shape parameter a. In view of the small sample sizes considered,

these results will perhaps be more useful in the actuarial ¬eld rather than in the

income distribution area.

We conclude this section by mentioning that Zandonatti (2001) recently suggested

a “generalized” Fisk distribution employing the procedure leading to Stoppa™s

generalized Pareto distribution. However, since a power transformation of the c.d.f.

leads to the c.d.f. of X1:n (or rather a generalization of its distribution with noninteger

n) and the order statistics of a Fisk parent follow a GB2 distribution [see (6.124)],

this approach does not lead to a “new” distribution.

6.4.2 Lomax (Pareto II) Distribution

A further two-parameter special case of the GB2 distribution, the Lomax

distribution”more precisely, the ¬rst Lomax distribution, since Lomax (1954)

introduced two distributions”has the c.d.f.

h xiÀq

, x . 0,

F(x) ¼ 1 À 1 þ (6:130)

b

227

6.4 FISK (LOG-LOGISTIC) AND LOMAX DISTRIBUTIONS

and the density

q x ÀqÀ1

x . 0:

f (x) ¼ 1 þ , (6:131)

b b

Hence, it is a Singh “ Maddala distribution with a ¼ 1. The Lomax distribution is

perhaps more widely known as the Pareto (II) distribution”this term is used, for

example, by Arnold (1983)”and is related to the classical Pareto distribution via