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Uniform Convergence of Smoothed Distribution Functions with an Application to Delta Method for the Lorenz Curve

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  • Kanaya, S.
  • Bhattacharya, D.

Abstract

In this note, we present some theoretical results useful for inference on a population Lorenz curve for income and expenditure distributions, when the population density of the distribution is not (uniformly) bounded away from zero, and potentially has thick tails. Our approach is to define Hadamard differentiability in a slightly nonstandard way, and using it to establish a functional delta method for the Lorenz map. Our differentiability concept is nonstandard in that the perturbation functions, which are used to compute the functional derivative, are assumed to satisfy certain limit conditions. These perturbation functions correspond to a (nonparametric) distribution function estimator. Therefore, as long as the employed estimator satis.es the same limit conditions, which we verify in this paper, the delta method and corresponding asymptotic distribution results can be established.

Suggested Citation

  • Kanaya, S. & Bhattacharya, D., 2017. "Uniform Convergence of Smoothed Distribution Functions with an Application to Delta Method for the Lorenz Curve," Cambridge Working Papers in Economics 1760, Faculty of Economics, University of Cambridge.
    • Handle: RePEc:cam:camdae:1760
    Note: db692
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    File URL: http://www.econ.cam.ac.uk/research-files/repec/cam/pdf/cwpe1760.pdf
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    References listed on IDEAS

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    1. Hansen, Bruce E., 2008. "Uniform Convergence Rates For Kernel Estimation With Dependent Data," Econometric Theory, Cambridge University Press, vol. 24(3), pages 726-748, June.
    2. Andrews, Donald W K, 1994. "Asymptotics for Semiparametric Econometric Models via Stochastic Equicontinuity," Econometrica, Econometric Society, vol. 62(1), pages 43-72, January.
    3. Bhattacharya, Debopam, 2007. "Inference on inequality from household survey data," Journal of Econometrics, Elsevier, vol. 137(2), pages 674-707, April.
    4. Kanaya, Shin, 2017. "Uniform Convergence Rates Of Kernel-Based Nonparametric Estimators For Continuous Time Diffusion Processes: A Damping Function Approach," Econometric Theory, Cambridge University Press, vol. 33(4), pages 874-914, August.
    5. Hirukawa, Masayuki & Sakudo, Mari, 2014. "Nonnegative bias reduction methods for density estimation using asymmetric kernels," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 112-123.
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    Cited by:

    1. Masayuki Hirukawa & Irina Murtazashvili & Artem Prokhorov, 2022. "Uniform convergence rates for nonparametric estimators smoothed by the beta kernel," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 1353-1382, September.

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