xm2 (1 þ x)Àm1 0x1

VI

(1 þ x2 )Àm exp (Àv arctan x) 21 , x , 1

IV

12

21 , x , 1

Normal exp À x

2

(1 À x2 )m 21 x 1

II

(1 þ x2 )Àm 21 , x , 1

VII

xm exp (Àx) 0 x,1

III

xÀm exp (À1=x) 0 x,1

V

(1 þ x)Àm

VIII 0x1

(1 þ x)m

IX 0x1

0 x,1

X exp (Àx)

xÀm 1 x,1

XI

[(g þ x)(g À x)]h 2g x g

XII

In applications the variable x is often replaced by (z À m)=s for greater ¬‚exibility.

The table contains several familiar distributions: Type I is the beta distribution of the

¬rst kind, type VI is the beta distribution of the second kind (with the F distribution

as a special case), type VII is a generalization of Student™s t, types III and V are the

gamma and inverse (or inverted, or reciprocal) gamma, respectively, type X is the

exponential, and type XI is the Pareto distribution.

A key feature of the Pearson system is that the ¬rst four moments (provided they

exist) may be expressed in terms of the four parameters a, c0 , c1 , c2 ; in turn, the

moment ratios

m2

b1 ¼ 3 (skewness)

m3

2

and

m4

b2 ¼ (kurtosis)

m2

2

provide a complete taxonomy of the Pearson curves. Indeed, Pearson suggested

selecting an appropriate density based on estimates of b1 , b2 that should then be

¬tted by his method of moments.

The main applications of the Pearson system are therefore in approximating

sampling distributions when only low-order moments are available and in providing

a family of reasonably typical non-Gaussian shapes that may be used, among other

things, in robustness studies. For further information on the Pearson distributions, we

51

2.3 SYSTEMS OF DISTRIBUTIONS

refer the reader to Johnson, Kotz, and Balakrishnan (1994, Chapter 12) or Ord

(1985) and the references therein.

In this book we mainly require the Pearson system for classifying size

distributions, several of which fall into this system (possibly after some simple

transformation). Speci¬cally, we shall encounter the Pearson type XI distribution

(under the more familiar name of Pareto distribution) in Chapter 3, the types III and

V (under the names gamma and inverse gamma distribution, respectively) in Chapter

5, and the type VI distribution (under the name beta distribution of the second kind)

in Chapter 6.

Of the many alternative systems of continuous univariate distributions, we will

also encounter some members of a system introduced by Irving Burr in 1942. Like

the Pearson system of distributions, the Burr family is de¬ned in terms of a

differential equation; unlike the Pearson system, this differential equation describes

the distribution function and not the density. This has the advantage of closed forms

for the c.d.f., sometimes even for the quantile function, which is rarely the case for

the members of the Pearson family. The Burr system comprises 12 distributions that

are usually referred to by number; see Table 2.4.

Table 2.4 The Burr Distributions

Type c.d.f. Support

0,x,1

I x

(1 þ eÀx )Àp À1 , x , 1

II

(1 þ xÀa )Àp 0,x,1

III

c À x1=c !Àq

0,x,c

IV 1þ

x

[1 þ c exp(À tan x)]Àq Àp=2 , x , p=2

V

[1 þ exp(Àc sinh x)]Àq À1 , x , 1

VI

2Àq (1 þ tanh x)q À1 , x , 1

VII

!q

2 x

À1 , x , 1

VIII arctan (e )

p

2

À1 , x , 1

IX 1À

2 þ c[(1 þ e x )q À 1]

[1 À exp(Àx2 )]a x,1

X 0

!q

1

0,x,1

XI xÀ sin(2px)

2p

1 À (1 þ xa )Àq x,1

XII 0

52 GENERAL PRINCIPLES

The c.d.f.™s of all Burr distributions satisfy the differential equation

F 0 (x) ¼ F(x)[1 À F(x)]g(x), (2:72)

where g is some nonnegative function.

The uniform distribution is clearly obtained for g ; [F(1 À F)]À1. The most

widely known of the (nonuniform) Burr distributions is the Burr XII distribution,

frequently just called the Burr distribution. In practice, one often introduces location

and scale parameters upon setting x ¼ (z À m)=s for additional ¬‚exibility. See

Kleiber (2003a) for a recent survey of the Burr family.

In Chapter 6 below we shall encounter the Burr III and Burr XII distributions,

albeit under the names of the Dagum and Singh “Maddala distributions.

Stoppa (1990a) proposed a further system of distributions that is closely related to

the Burr system. Rewriting the differential equation de¬ning the Burr distributions in

the form

F 0 (x)

¼ [1 À F(x)] Á g[x, F(x)], (2:73)

F(x)

˜

we see upon setting g(x, y) ¼: g(x, y)=x that Burr™s equation amounts to a