A discreet approach to study the distribution-free downward biases of Gini coefficient and the methods of correction in cases of small observations
It is well-known that Gini coefficient is influenced by granularity of measurements. When there are few observations only or when they get reduced due to grouping, standard measures exhibit a non-negligible downward bias. At times, bias may be positive when there is an apparent reduction in sample size. Although authors agreed on distribution-free and distribution-specific parts of it, there is no consensus in regard to types of bias, their magnitude and the methods of correction in the former. This paper deals with the distribution-free downward biases only, which arise in two forms. One is related to scale and occurs in all the cases stated above, when number of observations is small. Both occur together if initial number of observations is not sufficiently large and further they get reduced due to grouping. Underestimations associated with the former is demonstrated and addressed, for discontinuous case, through alternative formulation with simplicity following the principle of mean difference without repetition. Equivalences of it are also derived under the geometric and covariance approaches. However, when it arises with the other, a straightforward claim of it in its full magnitude may be unwarranted and quite paradoxical. Some exercises are done consequently to make Gini coefficient standardized and comparable for a fixed number of observations. Corrections in case of the latter are done accordingly with a newly proposed operational pursuit synchronizing the relevant previous and present concerns. The paper concludes after addressing some definitional issues in regard to convention and adjustments in cases of small observations.
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