p2 :

X1 !L X2 ( p1 (5:49)

)

Hence, the Lorenz order is linear within the family of two-parameter gamma

distributions.

Expressions for inequality measures are not as cumbersome as those of the

generalized gamma distribution. Speci¬cally, the Gini coef¬cient is given by

(McDonald and Jensen, 1979)

G( p þ 1=2)

p¬¬¬¬ :

G¼ (5:50)

G( p þ 1) p

The Pietra coef¬cient can be written in the two forms (McDonald and Jensen,

1979; Pham and Turkkan, 1994)

pp pp

1 1

P¼ 1 F1 (2; p þ 2; p) ¼ , (5:51)

e G( p þ 2) e G( p þ 1)

where 1 F1 is the con¬‚uent hypergeometric function.

Theil™s entropy measure T1 is given by (Salem and Mount, 1974)

1

þ c( p) À log p:

T1 ¼ (5:52)

p

As was to be expected from (5.49), all three coef¬cients are decreasing in p.

165

5.2 GAMMA DISTRIBUTION

5.2.5 Estimation

Regarding estimation, we shall again be brief since detailed accounts for the

estimation of two- and three-parameter gamma distributions are available in

Bowman and Shenton (1988) and Cohen and Whitten (1988), among other sources.

The likelihood equations for a simple random sample of size n are

X

n

^

log Xi À n log b À nc(^ ) ¼ 0,

p (5:53)

i¼1

X

n

p^

Xi À n^ b ¼ 0: (5:54)

i¼1

These can be solved iteratively, and indeed procedures for estimation in the gamma

distribution are nowadays available in many statistical software packages.

The Fisher information on u ¼ (b, p)` is given by

2p 3

1

6 b2 7

b

I (u) ¼ 6 7, (5:55)

41 5

c 0 ( p)

b

p¬¬¬ ^

from which the asymptotic covariance matrix of n(b, p)` can be obtained by

^

inversion or direct computation.

Equation (5.53) shows that the score function of the gamma distribution is

unbounded, implying that the MLE is very sensitive to outliers and other aberrant

observations. Victoria-Feser and Ronchetti (1994) and Cowell and Victoria-Feser

(1996) demonstrated by simulation for complete as well as truncated data that

parameter estimates and implied inequality measures for a gamma population can

indeed be severely biased when a nonrobust estimator such as the MLE is used.

They suggested employing robust methods such as an optimal bias-robust estimator

(OBRE) (see Section 3.6 for an outline of the basic ideas behind this estimator). If

only grouped data are available, Victoria-Feser and Ronchetti (1997) proposed a type

of minimum Hellinger distance estimator that enjoys better robustness properties

than the classical MLEs for grouped data.

McDonald and Jensen (1979) studied the sampling behavior of method of

moment estimators and MLEs of the Theil, Gini, and Pietra coef¬cients. They noted

that the knowledge of the sample arithmetic mean, geometric mean, and sample

variance are suf¬cient to calculate both the MLEs and method of moment estimators

of the Gini, Pietra, and Theil coef¬cients of inequality and provided a table that

facilitates these calculations.

The optimal grouping of data from a gamma population was considered by

Schader and Schmid (1986). The situation is somewhat more dif¬cult than in the

Pareto or lognormal cases (see Sections 3.6 and 4.6, respectively), since the optimal

class boundaries now depend on the shape parameter p and must therefore be

166 GAMMA-TYPE SIZE DISTRIBUTIONS

derived separately for each value of this parameter. Schader and Schmid reported

that there is always a unique set of optimal class boundaries in the gamma case.

Table 5.2 provides these boundaries zÃ , . . . , zÃÃ À1 based on the least number of

1 k

classes k Ã 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 and for p ¼ 0:5, 1, 2 and b ¼ 1. (See Section

3.6 for further details about this problem.)

From the table, optimal class boundaries aÃ for a gamma distribution with

j

parameters p and b can be obtained upon setting aÃ ¼ bzÃ .

j j

5.2.6 Empirical Results

Incomes and Wealth

March (1898) in his pioneering contribution ¬t the gamma model to wage

distributions for France, Germany, and the United States, strati¬ed by occupation.

Salem and Mount (1974) applied the gamma distribution to U.S. pre-tax personal

incomes for 1960 “ 1969, concluding that the gamma provides a better ¬t than the

lognormal distribution. Speci¬cally, the Gini coef¬cients implied by the estimated

gamma distributions mostly fall within the feasible bounds de¬ned by Gastwirth

(1972), whereas those implied by lognormal ¬ttings do not.

Kloek and van Dijk (1977) employed the gamma distribution when analyzing

data originating from the Australian survey of consumer expenditures and ¬nances

Table 5.2 Optimal Class Boundaries for Gamma Data

kÃ zÃ ; . . . ; zÃÃ À1

g

p 1 k

0.5 0.1 5 0.0005 0.0127 0.1002 0.5119

0.05 7 0.0001 0.0021 0.0161 0.0742 0.2603 0.8293

0.025 10 0.0000 0.0003 0.0023 0.0102 0.0342 0.0946 0.2313 0.5313

1.2486

0.01 16 0.0000 0.0000 0.0002 0.0007 0.0024 0.0065 0.0153 0.0325

0.0638 0.1179 0.2083 0.3574 0.6057 1.0401 1.9170

1 0.1 5 0.0351 0.1792 0.5416 1.4073

0.05 7 0.0143 0.0712 0.2036 0.4607 0.9398 1.9215

0.025 10 0.0054 0.0265 0.0742 0.1611 0.3041 0.5289 0.8804 1.4543

2.5316

0.01 17 0.0012 0.0059 0.0164 0.0350 0.0643 0.1070 0.1665 0.2468

0.3531 0.4923 0.6744 0.9148 1.2370 1.6872 2.3615 3.5437

2 0.1 5 0.3479 0.8515 1.6372 3.0202

0.05 7 0.2174 0.5113 0.9195 1.4884 2.3246 3.7469

0.025 10 0.1314 0.3017 0.5241 0.8996 1.1640 1.6233 2.2347 3.1071

4.5599

0.01 17 0.0615 0.1386 0.2350 0.3505 0.4859 0.6428 0.8238 1.0326

1.2746 1.5574 1.8920 2.2956 2.7965 3.4460 4.3535 5.8377

Source: Schader and Schmid (1986).

167

5.2 GAMMA DISTRIBUTION

for the period 1966 “ 1968 (see Podder, 1972). Here the ¬t is not impressive, the