Stoppa then proposed a differential equation for the elasticity

1 À [F(x)]1=u

h(x, F) ¼ x ! x0 . 0,

Á g[x, F(x)], (2:74)

[F(x)]1=u

where u . 0, and g(x, y) is positive in 0 , y , 1.

For g(x, y) ¼ g(x), F(x) = 0, 1, and dF 1=u =dF ¼ F 1À1=u =u, the solution of this

differential equation is

& ° 1 '

˜(t) dt

F(x) ¼ 1 À exp ,

0

where ˜(t) ¼ g(t)=(ut) is a real function integrable on a subset of Rþ . For example,

if g is chosen as bux=(1 À bx), b, bx, 1=(b À x), bx sec2 x, respectively, with b . 0,

we obtain the following c.d.f.™s:

F(x) ¼ (bx)u 0 , x , 1=b

I:

Àb u

1,x,1

F(x) ¼ (1 À x )

II:

F(x) ¼ (1 À eÀbx )u 0,x,1

III:

F(x) ¼ [1 À (bxÀ1 À 1)1=(ub) ]u b=2 , x , b

IV:

F(x) ¼ (1 À beÀ tan x )u Àp=2 , x , p=2

V:

53

2.3 SYSTEMS OF DISTRIBUTIONS

For u ¼ 1 we arrive at various cases of the Burr system, whereas for u = 1 type I

de¬nes a power function distribution, type II a generalized Pareto, and type III a

generalized exponential distribution, respectively.

In a later paper, Stoppa (1993) presented a classi¬cation of distributions inspired

by the classical table of chemical elements of Mendeleyev. Stoppa™s table comprises

15 so-called periods de¬ned by families of distributions for which log h(x, F)

depends on a single parameter. Within each period there are subfamilies of

distributions characterized by up to ¬ve parameters.

As far as income distributions are concerned, special interest is focused on family

no. 31 of the system whose c.d.f. is given by

& b3 'À1=b1

x

þc , (2:75)

Àb1

b3

where in general b1 = 0 and b3 = 0. For b1 , 0, b3 , 0, we get the generalized

Pareto type I (the Stoppa distribution), for b1 . 0, b3 , 0, we obtain the inverted

Stoppa distribution; b1 , 0, b3 . 0 yields a generalized power function; b1 . 0,

b3 . 0 gives us the Burr III (Dagum type I) distribution; and ¬nally for b1 ¼ À1,

b3 , 0, we obtain the classical Pareto type I distribution.

Transformation Systems

The Pearson curves were designed in such a manner that for any possible pairs of

p¬¬¬¬¬

values b1 , b2 there is just one corresponding member of the Pearson family of

distributions. Alternatively, one may be interested¬¬¬¬¬ a transformation, to normality,

p in

say, such that for any possible pairs of values b1 , b2 there is one corresponding

normal distribution. Unfortunately, no such single transformation is available;

however, Johnson (1949) has described a set of three transformations which, when

p¬¬¬¬¬

combined, do provide one distribution corresponding to each pair of values b1 , b2 .

These transformations are

Z ¼ m þ s log(X À l), X . l, (2:76)

& '

X Àl

Z ¼ m þ s log l , X , l þ j,

, (2:77)

lþjÀX

and

& '

X Àl

Z ¼ m þ s sinhÀ1 À 1 , X , 1:

, (2:78)

j

Here Z follows a standard normal distribution and m, s, l, j represent parameters of

which j must be positive and s non-negative.

54 GENERAL PRINCIPLES

The distributions de¬ned by the preceding equation are usually denoted as SL , SB ,

and SU . Below we shall deal only with the SL distributions, under the more familiar

name of three-parameter lognormal distributions.

It is natural to extend Johnson™s approach to nonnormal random variables Z. For a

Z following a standard logistic distribution with c.d.f.

1

À1 , z , 1,

F(z) ¼ ,

1 þ eÀz

the distributions associated with the corresponding sets of transformations have been

described by Tadikamalla and Johnson (1982) and Johnson and Tadikamalla (1992).

The resulting distributions are usually denoted as LL , LB , and LU . For a Z following a

Laplace (or double exponential) distribution with p.d.f.

1

f (z) ¼ eÀjzj , À1 , z , 1,

2

the corresponding distributions have been discussed by Johnson (1954); they are

0 0 0

denoted as SL , SB ; and SU .

We shall encounter distributions of the LL type”that is, log-logistic distributions”

0

and generalizations thereof in Chapter 6 and (generalizations of) SL distributions in

Chapter 4, albeit under the name of generalized lognormal distributions. In fact, with

little exaggeration, this book can be considered a monograph on exponential

transformations of some of the more familiar statistical distributions. Speci¬cally,

if Z is

Exponential, then exp (Z) follows a Pareto distribution”a distribution studied

.

in Chapter 3.

Normal, then exp (Z) follows, of course, a lognormal distribution”a distri-

.

bution studied in Chapter 4.

Gamma, then exp (Z) follows a loggamma distribution”a distribution studied

.

in Chapter 5.

Logistic, then exp (Z) follows a log-logistic distribution”a distribution studied,

.

along with its generalizations, in Chapter 6.

Rayleigh, then exp (Z) follows a Benini distribution”a distribution studied in

.

Chapter 7.

Less prominent choices for Z include the exponential power (Box and Tiao, 1973)

and the Perks (1932) distributions, their exponential siblings are called generalized

lognormal and Champernowne distributions and are explored in Sections 4.10 and

7.3, respectively. Very few distributions studied here do not originate from an

exponential transformation, mainly the gamma-type models of Chapter 5.

55