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Interpolating the L orenz Curve: Methods to Preserve Shape and Remain Consistent with the Concentration Curves for Components
Author(s) -
Okamoto Masato
Publication year - 2014
Publication title -
review of income and wealth
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.024
H-Index - 57
eISSN - 1475-4991
pISSN - 0034-6586
DOI - 10.1111/roiw.12083
Subject(s) - convexity , interpolation (computer graphics) , piecewise , parametric statistics , mathematics , parametric equation , class (philosophy) , monotonic function , boundary (topology) , parametric model , data point , curve fitting , mathematical optimization , computer science , statistics , geometry , mathematical analysis , economics , artificial intelligence , motion (physics) , financial economics
C 1 ‐class interpolation methods that preserve monotonicity and convexity and are thus suitable for the estimation of the L orenz curve from grouped data are not widely known. Instead, parametric models are usually applied for such estimation. Parametric models, however, have difficulty in accurately approximating every part of income/expenditure distributions. This paper proposes two types of C 1 ‐class shape‐preserving interpolation methods. One is a piecewise rational polynomial interpolation (proposed independently by S tineman and D elbourgo) that enables consistent interpolation of the concentration curves for income/expenditure components, attaining approximately the same accuracy as that of the existing methods when applied to decile‐grouped data or to more detailed aggregation. Another is a H ybrid interpolation that employs pieces of curves derived from parametric models on end intervals. Empirical comparisons show that the H ybrid interpolation (with the assistance of parametric models for class‐boundary estimation) outperforms the existing methods even when applied to quintile‐grouped data without class boundaries.