Hogg and Klugman (1983) ¬t the Lomax distribution (under the name of Pareto

distribution) to data on malpractice losses, for which it is preferable over the beta II,

Singh “Maddala, lognormal, and Weibull distributions. The data require a very

heavy-tailed model with a parameter q slightly below 1.

Cummins et al. (1990) applied the distribution to two sets of ¬re liability data but

the performance of the Lomax distribution is not impressive. It ranks only 12th and

13th out of 16 distributions of the gamma and beta type.

In a recent investigation studying losses from catastrophic events in the United

States, Burnecki, Kukla, and Weron (2000) employed a Pareto type II distribution

and obtained a tail index q in the vicinity of 2.7.

In summary, it would seem that most data on size distributions require a more

¬‚exible distribution than the Fisk or Lomax distributions. Speci¬cally, an additional

shape parameter appears to be appropriate.

6.5 (GENERALIZED) BETA DISTRIBUTION OF THE

FIRST KIND

Occasionally, distributions supported on a bounded domain have been considered for

the modeling of size phenomena, notably the distribution of income. The most

¬‚exible of these are the generalized beta distribution of the ¬rst kind (hereafter

referred to as GB1) and its special case, the standard beta distribution. We refer to

Johnson, Kotz, and Balakrishnan (1995, Chapter 25) for the basic properties of this

well-known distribution, and we shall brie¬‚y mention below some aspects pertaining

to size phenomena. The GB1 was introduced by McDonald (1984) as an income

distribution, whereas the B1 was used for the same purpose more than a decade

earlier by Thurow (1970).

6.5.1 De¬nition and Properties

The GB1 is de¬ned in terms of its density

axapÀ1 [1 À (x=b)a ]qÀ1

f (x) ¼ , 0 x b, (6:137)

bap B( p, q)

where all four parameters a, b, p, q are positive. Here b is a scale and a, p, q are

shape parameters. When a ¼ 1, we get

xpÀ1 [1 À x=b]qÀ1

f (x) ¼ , 0 x b, (6:138)

bp B( p, q)

the three-parameter beta distribution.

231

6.5 (GENERALIZED) BETA DISTRIBUTION OF THE FIRST KIND

The GB1 is related to the GB2 distribution via the relation

1=a

Xa

X $ GB2(a, b, p, q) ¼ $ GB1(a, b, p, q):

)

1 þ Xa

This generalizes a well-known relationship between the B1 and B2 distributions.

The c.d.f.™s of the GB1 and B1 distributions cannot be expressed in terms of

elementary functions. However, in view of (6.8), they are available in terms of

Gauss™s hypergeometric function 2 F1 , in the form (McDonald, 1984)

h x a i

(x=b)a

:

F(x) ¼ 2 F1 p, 1 À q; p þ 1; (6:139)

pB( p, q) b

In analogy with the GB2 case discussed in Section 6.1, they can also be written as an

incomplete beta function ratio

x a

F(x) ¼ Iz ( p, q), where z ¼ , (6:140)

b

in the GB1 case (of course, a ¼ 1 yields the c.d.f. of the three-parameter B1

distribution).

The moments of the GB1 exist for Àap , k , 1; they are

bk B( p þ k=a, q) bk G( p þ k=a)G( p þ q)

k

:

E(X ) ¼ (6:141)

¼

G( p þ q þ k=a)G( p)

B( p, q)

An analysis of the hazard rate is more involved than for either the generalized

gamma or T GB2 distribution. Monotonically decreasing, monotonically

the S

increasing, as well as -shapes are possible; see McDonald and Richards

(1987) for a discussion of these possibilities.

The Lorenz ordering within the GB1 family was studied by Wil¬‚ing (1996c),

who provided four sets of suf¬cient conditions. Noting that the GB1 density is

regularly varying at the origin with index Àap À 1, it can be deduced along the lines

of Kleiber (1999b, 2000a) that a1 p1 a2 p2 is a necessary condition for X1 !L X2. A

complete characterization of the Lorenz ordering within this family appears to be

unavailable at present.

However, Sarabia, Castillo, and Slottje (2002) provided Lorenz ordering

results for nonnested income distributions. These include the following: That for

X $ GB2(a, b, p, q) and Y $ GB1(a, b, p, q), one has X !L Y , and for X $

GB1(a, b, p, q) and Y following a generalized gamma distribution (cf. Chapter 5),

a˜˜ ˜ ˜˜

that is, Y $ GG(˜ , b, p) with a ! a and ap ! ap, it follows that Y !L X .

McDonald (1984) provided the Gini coef¬cient of the GB1 as a somewhat

lengthy expression involving the generalized hypergeometric function 4 F3 . For the

232 BETA-TYPE SIZE DISTRIBUTIONS

B1 subfamily the expression is less cumbersome and equals (McDonald and

Ransom, 1979a)

G( p þ q)G( p þ 1=2)G(q þ 1=2)

2B( p þ q, 1=2)B( p þ 1=2, 1=2)

¼ p¬¬¬¬ :

G¼

pB(q, 1=2) pG( p þ q þ 1=2)G( p þ 1)G(q)

The Pietra index and the (¬rst) Theil coef¬cient of the B1 distribution are

(McDonald, 1981; Pham-Gia and Turkkan, 1992)

[ p=( p þ q)]p p

P¼ 2 F1 p, 1 À q, p þ 2;

p( p þ 1)B( p, q) pþq

and

!

p

T1 (X ) ¼ c ( p þ 1) À c ( p þ q þ 1) À log ,

pþq

respectively.

A distribution encompassing both the GB1 and GB2 families was proposed

by McDonald and Xu (1995). This ¬ve-parameter generalized beta distribution has

the p.d.f.

jajxapÀ1 [1 þ (1 À c)(x=b)a ]qÀ1

f (x) ¼ ap , (6:142)

b B( p, q)[1 þ c(x=b)a ]pþq