Bounds of the Gini Index Using Sparse Information on Mean Incomes

Confidentiality, political sensitivity and other considerations often dictate the reporting of income inequality data by governments and other agencies in grouped format. In these cases, inequality measures, such as the Gini index, must be estimated from the grouped data rather than from the individual observations from which the grouped data are constructed. Also, grouped data, which are generally more widely available than micro-data, are particularly useful for cross-country studies of income inequality. Several approaches to the measurement of the Gini index based on grouped data have been adopted in the literature. One parametric approach entails fitting a theoretical Lorenz curve (LC) to the grouped data and the Gini index (and in some cases the underlying density) is then deduced from the estimated parameters of the LC. See, for example, Kakwani and Podder (1973, 1976), Villaseñor and Arnold (1989), Ogwang and Rao (1996), Sarabia (1997), and Sarabia et al. (1999). Schader and Schmid (1994) have provided a fairly comprehensive survey of more than ten parametric LCs. Another parametric approach applies interpolation methods to deduce estimates of the Gini index, assuming the observed points on the LC to be fixed. See, for example, Gastwirth (1975), Gastwirth and Glauberman (1976), Cowell and Mehta (1982), and Brown and Mazzarino (1984). More than two decades ago, Gastwirth (1972), Mehran (1975), Murray (1978), and Fuller (1979) proposed non-parametric approaches by deriving lower and upper bounds, from grouped data, within which the Gini index must lie regardless of the functional form of the underlying distribution of income. The appeal of these non-parametric approaches stems from the fact that no assumption regarding the shape of the underlying LC (or the corresponding income distribution) is necessary. The lower bound assumes that the incomes in each bracket are equally distributed whereas the upper bound also incorporates a “grouping