xkþ1 ¼ 1 be the boundaries of the k þ 1 groups and denote the number of

observations, in a random sample of size n, falling into the interval [xi , xiþ1 ) by ni

P

( k ni ¼ n). The MLE of a is then the solution of

i¼0

X xÀa log xiþ1 À xÀa log xi

k À1

ni iþ1 i

þ nk log xk ¼ 0: (3:69)

Àa À xÀa

xiþ1 i

i¼0

82 PARETO DISTRIBUTIONS

It is worth noting that if the group boundaries form a geometric progression, with

xiþ1 ¼ cxi , i ¼ 1, . . . , k, then the MLE can be expressed in closed form, namely,

!

1 n

^

a¼ :

log 1 þ Pk (3:70)

log c ini

i¼0

3.6.3 Optimal Grouping

The prevalence of grouped data in connection with the distribution of income or

wealth leads naturally to the question of optimal groupings. Only fairly recently,

Schader and Schmid (1986) have addressed this problem in a likelihood framework.

Suppose that a sample X ¼ (X1 , . . . , Xn )` of size n is available and the parameter

of interest is a. (Clearly, this is the relevant parameter in connection with inequality

measurement as the formulas in Section 3.5 indicate.) By independence, the Fisher

information of the sample on a is

n

I (a) ¼ nI1 (a) ¼ ,

a2

where I1 denotes the information in a single observation. For a given number of

groups k with group boundaries X0 ¼ a0 , a1 , Á Á Á , akÀ1 , ak ¼ 1, de¬ne the

class frequencies Nj as the number of Xi in [ajÀ1 , aj ), j ¼ 1, . . . , k. Thus, the joint

distribution of N ¼ (N1 , . . . , Nk )` is multinomial with parameters n and pj ¼ pj (a),

where

° aj

f (x j a) dx, j ¼ 1, . . . , k:

pj (a) ¼

ajÀ1

Now the Fisher information in N is

X [@pj (a)=@a]2 n X [ log (zj )zj À log (zjÀ1 )zjÀ1 ]2

k k

IN (a) ¼ n ,

¼2

a i¼1

pj (a) zjÀ1 À zj

i¼1

where z0 ¼ 1, zk ¼ 0, zj ¼ (aj =x0 )Àa , j ¼ 1, . . . , k À 1. This expression is, for ¬xed

a, a function of k and the k À 1 class boundaries a1 , . . . , akÀ1 .

Passing from the complete data X to the class frequencies N implies a loss of

information that may be expressed in terms of the decomposition

IX (a) ¼ IN (a) þ IX jN (a):

The relative loss of information is then given by

L ¼ 1 À a2 IN (a):

83

3.6 ESTIMATION

L is a function of k, a, and a1 , . . . , akÀ1, but not a function of n. Given k and a, the

loss of information is now minimized if and only if the class boundaries aÃ , . . . , aÃ

1 kÀ1

are de¬ned by

IN (a; aÃ , . . . , aÃ ) ¼ IN (a; a1 , . . . , akÀ1 ):

sup

1 kÀ1

aÃ ,ÁÁÁ,aÃ

1 kÀ1

The boundaries aÃ , . . . , aÃ are now called optimal class boundaries and the

1 kÀ1

corresponding intervals [aÃ , aÃ ), j ¼ 1, . . . , k, are called an optimal grouping. By

jÀ1 j

using these boundaries, the number of classes k may now be determined in such a

way that the loss of information due to grouping does not exceed a given bound, g,

say. Thus, one requires, for given g [ (0, 1) and a, the smallest integer k Ã such that

L(a; k Ã ; aÃ , . . . , aÃ ) g:

1 kÀ1

The relevant k can now be found by determining the boundaries and the information

loss for k ¼ 1, 2, . . . until for the ¬rst time L is less than or equal to g. In practice,

the parameter a will remain unknown and have to be replaced by an estimate.

Schader and Schmid (1986) argued that, although the corresponding class

boundaries will not be optimal in that case, the loss of information will often be

less than using ad hoc determined class boundaries.

Table 3.1 provides optimal class boundaries zÃ based on the least number

j

Ã

of classes k for which the loss of information is less than or equal to a given value of

g, for g ¼ 0:1, 0.05, 0.025, and 0.01. From the table, optimal class boundaries aÃ j

for a Pareto distribution with parameters a and x0 can be obtained by setting

aÃ ¼ x0 zÃÀ1=a .

j

j

3.6.4 Unbiased Estimation of Pareto Characteristics

Over the last 10“ 15 years there has been considerable interest in the UMVU

estimation of various Pareto characteristics, notably the parameters, the density, the

c.d.f., the moments, and several inequality indices.

An improvement over the MLEs is obtained by removing their biases. In the case

^

where both parameters are unknown, this amounts to replacing x0 by the estimator

& '

1

xÃ ¼ X1:n 1 À , (3:71)

0

^

(n À 1)a

Table 3.1 Optimal Class Boundaries for Pareto Data

kÃ zÃ , . . . , zÃÃ À1

g 1 k

0.1 5 0.5486 0.2581 0.0933 0.0190

0.05 7 0.6521 0.3958 0.2171 0.1021 0.0369 0.0075

0.025 9 0.7173 0.4935 0.3218 0.1953 0.1071 0.0504 0.0182

0.01 15 0.8192 0.6616 0.5256 0.4097 0.3122 0.2315 0.1660

0.1142 0.0745 0.0452 0.0248 0.0117 0.0042 0.0009

Source: Schader and Schmid (1986).

84 PARETO DISTRIBUTIONS

^

and replacing a by