(1983).

As an alternative to ML estimation, the maximum product of spacings (MPS)

estimation was considered by Shah and Gokhale (1993). This method obtains

estimates of a vector-valued parameter u by maximizing

1X nþ1

log{F(xi , u) À F(xiÀ1 , u)},

H¼

n þ 1 i¼1

i ¼ 1, 2, . . . , n þ 1, with x0 ¼ À1 and xnþ1 ¼ 1. From a simulation study

employing ten parameter combinations and nine sample sizes ranging from n ¼ 10

to n ¼ 150, Shah and Gokhale concluded that MPS is superior to ML estimation, in

the sense of smaller MSE, at least for small samples.

209

6.2 SINGH “ MADDALA DISTRIBUTION

6.2.6 Empirical Results

Incomes and Wealth

Singh and Maddala (1976) compared their model to Salem and Mount™s (1974) results

for the gamma distribution using 1960“ 1972 U.S. family incomes and concluded that

their model provides a better ¬t than either the gamma or lognormal functions.

However, Cronin (1979), in a comment on the Singh and Maddala paper, observed

that the implied Gini indices for the Singh “ Maddala model almost always fall outside

the Gastwirth (1972) bounds (calculated by Salem and Mount, 1974) for their data.

He concluded that Singh and Maddala™s claim that the “Burr distribution ¬ts the data

better [than the gamma distribution] would now appear to be questionable” (p. 774).

When compared to lognormal, gamma, and beta type I ¬ttings for U.S. family

incomes for 1960 and 1969 through 1975, the Singh “ Maddala distribution generally

outperforms all these distributions, with only the beta type I being slightly better in a

few cases (McDonald and Ransom, 1979a).

Dagum (1983) ¬t a Singh “ Maddala distribution to 1978 U.S. family incomes for

which it outperforms the (two-parameter) lognormal and gamma distributions by

wide margins. However, the (four- and three-parameter) Dagum type III and type I

distributions ¬t even better.

In McDonald (1984) the distribution ranks second out of 11 considered models”

being inferior only to the GB2 distribution”when ¬tted to 1970, 1975, and 1980

U.S. family incomes.

For Japanese incomes for 1963 “1971, the distribution outperforms the Fisk, beta,

gamma, lognormal, and Pareto (II) distribution. However, a likelihood ratio test

reveals that the full ¬‚exibility of the Singh “ Maddala distribution is not required and

that the two-parameter Fisk distribution already provides an adequate ¬t (Suruga,

1982). The distribution was also ¬tted to various strata from the 1975 Japanese

Income Redistribution Survey by Atoda, Suruga, and Tachibanaki (1988). Four

occupational classes as well as primary and redistributed incomes were considered,

and ¬ve different estimation techniques were applied. In a later study using the same

data set, Tachibanaki, Suruga, and Atoda (1997) considered ML estimators on the

basis of individual observations. Here the Singh “Maddala model is almost always

the best out of six different functions [including (generalized) gamma and Weibull

distributions] in terms of several ¬t criteria. However, when the AIC is employed for

model selection, the Singh “ Maddala distribution turns out to be essentially

overparameterized for one stratum, with a two-parameter log-logistic special case

providing an adequate ¬t.

Henniger and Schmitz (1989) employed the Singh “Maddala distribution when

¬tting ¬ve parametric models to data from the UK Family Expenditure Survey for

1968 “1983. However, for the whole population all parametric models are rejected;

for subgroups the Singh “ Maddala distribution performs better than any other

parametric model considered and appears to be adequate for their data, in terms of

goodness-of-¬t tests.

Majumder and Chakravarty (1990) considered the Singh “ Maddala distribution

when modeling U.S. income data from 1960, 1969, 1980, and 1983. The

210 BETA-TYPE SIZE DISTRIBUTIONS

distribution is among the best three-parameter models. In a reassessment of

Majumder and Chakravarty™s ¬ndings, McDonald and Mantrala (1993, 1995) ¬t

the Singh “ Maddala distribution to 1970, 1980, and 1990 U.S. family incomes,

using two different ¬tting methods and alternative groupings of the data. Here the

distribution is outperformed by the more ¬‚exible GB2 as well as the Dagum

distribution.

McDonald and Xu (1995) studied 1985 U.S. family incomes; out of 11 distri-

butions of the beta and gamma type, the Singh “ Maddala ranks fourth in terms of

likelihood and several other goodness-of-¬t criteria.

In an application to 1984 “1993 German household incomes, the Singh “ Maddala

distribution emerges as one of two suitable models (Brachmann, Stich, and

Trede, 1996), being comparable to the more general GB2 distribution. Two-

parameter models such as the gamma or Weibull distributions are not appropriate for

these data.

Bordley, McDonald, and Mantrala (1996) ¬t the Singh “ Maddala distribution to

U.S. family incomes for 1970, 1975, 1980, 1985, and 1990. The distribution ranks

fourth out of 15 considered models of the beta and gamma type, being outperformed

only by the GB2, GB, and Dagum type I distributions and improving on three- and

four-parameter models such as the generalized gamma and GB1 distributions. For

1985 the relative ranking of the Singh “ Maddala and Dagum distributions depends

on the criterion selected; in terms of likelihood and x 2, the Singh “ Maddala does

slightly better.

Bell, Klonner, and Moos (1999) ¬t the Singh “ Maddala distribution to the per

capita consumption expenditure data of rural Indian households for 28 survey

periods, stretching from 1954 “ 1955 to 1993“ 1994. They reported that for a total of

44.4% of all 378 possible pairs, a ranking in terms of ¬rst-order stochastic

dominance is possible, and another 47.7% can be Lorenz-ordered.

Botargues and Petrecolla (1997) estimated the Singh “Maddala model for the

income distribution in the Buenos Aires region, for the years 1990 “ 1996. However,

the Dagum distributions (of various types) perform better on these data.

Actuarial Losses

In the actuarial literature, Hogg and Klugman (1983) ¬t a Singh “ Maddala

distribution (under the name of Burr distribution) to 35 observations on hurricane

losses in the United States. Compared to the Weibull and lognormal models, the

distribution appears to be overparameterized for these data.

Cummins et al. (1990), in their hybrid paper already mentioned above, ¬t the

Singh “Maddala distribution (under the name of Burr XII) to aggregate ¬re losses

(the data are provided in Cummins and Freifelder, 1978). The distribution performs

quite well; however, less generously parameterized limiting forms of the GB2 such

as the inverse exponential distribution seem to do even better. The same authors also

considered data on the severity of ¬re losses and ¬t the GB2 to both grouped and

individual observations. Here the Singh “ Maddala is indistinguishable from the four-

parameter GB2 distribution for individual observations and is therefore a distribution

of choice for these data.

211

6.2 SINGH “ MADDALA DISTRIBUTION

6.2.7 Extensions

A generalization of the Singh “ Maddala distribution was recently proposed by

Zandonatti (2001). Using Stoppa™s (1990a,b) method leading to a power

transformation of the c.d.f., the resulting density is given by

aquxaÀ1 h xa iÀqÀ1 n h xa iÀq ouÀ1

, x . 0:

f (x) ¼ 1þ 1À 1þ (6:81)

ba b b

Clearly, for u ¼ 1 we arrive at the Singh “ Maddala distribution.

More than a decade before the publication of Singh and Maddala™s pioneering

1976 paper, a multivariate Singh “ Maddala distribution was proposed by Takahasi

(1965), under the name of multivariate Burr distribution. This distribution is de¬ned

by the joint survival function

( )Àq

X xi ai

k

F (x1 , . . . , xk ) ¼ 1 þ xi . 0, i ¼ 1, . . . , k,

, (6:82)

bi

i¼1

and is often referred to as the Takahasi “Burr distribution. For the components Xi of

a random vector (X1 , . . . , Xk ) following this distribution, we have the representation

1=ai

Yi

d

, i ¼ 1, . . . , k,

Xi ¼ bi (6:83)

Z

where the Yi ™s are i.i.d. following a standard exponential distribution and Z has a