quantile-type estimators for all sample sizes under consideration. For instance, if

n ! 25 and protection against 10% upper outliers is considered suf¬cient, a(2) with

^ GM

k ¼ 4 yields a relative ef¬ciency of more than 0.89 and a breakdown point !0:12.

3.6.6 Miscellaneous Estimators

Kang and Cho (1997) derived the MSE-optimal estimator of the shape parameter a,

for unknown scale x0 , within the class of estimators of the form c= log(Xj =^ 0 ) [note

x

that both the MLE (3.62) and UMVUE (3.72) are of this form] obtaining

nÀ3

a ¼ Pn

˜ (3:91)

i¼1 log(Xi =^ 0 )

x

with an MSE of

a2

˜

MSE(a) ¼ , n . 2: (3:92)

nÀ2

They referred to this estimator as the minimum risk estimator (MRE). Compared to

the MLE (3.62), it not only has smaller MSE but a smaller bias as well

a

˜

bias(a) ¼ À (3:93)

nÀ2

(half of the bias of the MLE, in absolute value). Kang and Cho also studied

jackknifed and bootstrapped versions of the MLEs and MREs when one of the

parameters is known.

In Section 3.4 we saw that the Pareto distribution can be characterized in terms of

maximum entropy, and it is also possible to obtain estimators of its parameters based

on this principle. This leads to the estimating equations

s¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

1

x0 ¼ (3:94)

var( log X )

and

& '

1

a ¼ exp E( log X ) À : (3:95)

x0

Singh and Guo (1995) have studied the performance of these estimators in relation to

the MLEs and two further estimators by Monte Carlo. It turns out that the maximum

entropy estimators perform about as well as the MLEs in terms of bias and root mean

square error, and often much better than the method of moments estimators,

especially in small samples.

90 PARETO DISTRIBUTIONS

3.6.7 Inequality Measures

The maximum likelihood estimators of the Lorenz curve (3.51), Gini index (3.53),

and Theil index (3.56) are obtained by replacing the shape parameter a by its

^

estimator a depending on whether x0 is known or unknown and then introducing the

condition L(u) ! 0.

Moothathu (1985) has studied the sampling distributions of the ML estimators of

L(u) and the Gini index, which are of the mixed type, and shown that the estimators

are strongly consistent. In the case of the Gini coef¬cient, the sampling distribution

^

of the MLE possesses the density fG (w) ¼ p þ (1 À p) f (w), where p ¼ P(G ! 1) ¼

^

^

P(a 1) and

nÀs

(2na)nÀsþ1 w À2naw

(1 þ w)À2 exp 0 , w , 1, (3:96)

f (w) ¼ ,

G(n À s þ 1) 1 þ w 1þw

with s ¼ 1(2) if x0 is known (unknown).

UMVUEs of the Lorenz curve, Gini index, and Theil coef¬cient were derived by

Moothathu (1990c). [The UMVUE of the Gini coef¬cient for both parameters

unknown may already be found in Arnold (1983, p. 200).] In what follows,

Si , i ¼ 1, 2, are de¬ned as in (3.77) and (3.78). For known x0 , an unbiased estimator

of L(u) is

^

LÃ ¼ 1 À (1 À u) 0 F1 [À; n; ÀnS1 log(1 À u)]

X {ÀnS1 log(1 À u)}j

1

¼ 1 À (1 À u) , (3:97)

(n)j j!

j¼0

where 0 F1 is a Bessel function and (n)j is Pochhammer™s symbol for the forward

factorial function (see p. 287 for a de¬nition), and in the case where x0 is unknown,

the UMVUE is

^

LÃ ¼ 1 À (1 À u) 0 F1 [À; n À 1; ÀnS2 log(1 À u)]: (3:98)

The UMVUEs of the Gini coef¬cient are

nS1

1 F1 1; n; À1 if x0 is known (3:99)

2

nS2

1 F1 1; n À 1; À 1 if x0 is unknown, (3:100)

2

where 1 F1 is Kummer™s con¬‚uent hypergeometric function.

91

3.7 EMPIRICAL RESULTS

The UMVUEs of the coef¬cient of variation were given in Section 3.6.4 above.

For the Theil coef¬cient, the UMVUEs are

2

nS1

2 F2 (2, 2; 3, n þ 2; nS1 ) if x0 is known (3:101)

2(n þ 1)

2

nS2

2 F2 (2, 2; 3, n þ 1; nS2 ) if x0 is unknown, (3:102)

2(n À 1)

where 2 F2 is a generalized hypergeometric function (see p. 288 for a de¬nition).

Ali et al. (2001) showed that the UMVUE of the Gini coef¬cient has a variance in

the vicinity of the Cramer“ Rao bound and is considerably more ef¬cient than the

´

MLE in terms of MSE, in small samples. The same holds true for the Lorenz curve

for large quantiles.

Latorre (1987) derived the asymptotic distributions of the ML estimators of the

Gini coef¬cient and of Zenga™s measures j and j2, and the sampling distributions of

^

the latter indices were obtained by Stoppa (1994). Since P(a 1) . 0 although the

coef¬cients only exist for a . 1, these distributions are all of the mixed type.

3.7 EMPIRICAL RESULTS

Having already sketched the early history of income distributions in Chapter 1 we

shall here con¬ne ourselves to the comparatively few more recent studies employing

Pareto distributions in that area and add some material on the distribution of wealth

and on the size distributions of ¬rms and insurance losses.

Income Data

In a reexamination of Pareto™s (1896, 1897b) results for the Pareto type II

distribution, Creedy (1977) noted some inconsistencies in Pareto™s statements. He