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Shape-restricted inference for Lorenz curves using duality theory

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  • Dentcheva, Darinka
  • Penev, Spiridon

Abstract

We propose a new methodology for estimating curves under a convexity restriction based on Fenchel duality and wavelet approximations. In contrast to approaches where a possibly non-convex estimator is convexified at a second stage, our procedure allows us to construct directly an estimator with a convex shape. The method is applied to the estimation of the Lorenz curve. Applications to estimation of average value at risk, as well as multivariate generalisations to Lorenz surfaces are mentioned. We show asymptotic efficiency which demonstrates that the convexity is achieved at no extra cost.

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  • Dentcheva, Darinka & Penev, Spiridon, 2010. "Shape-restricted inference for Lorenz curves using duality theory," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 403-412, March.
    • Handle: RePEc:eee:stapro:v:80:y:2010:i:5-6:p:403-412
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    References listed on IDEAS

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    1. Andrzej Ruszczyński & Alexander Shapiro, 2006. "Optimization of Convex Risk Functions," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 433-452, August.
    2. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
    3. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
    4. Melanie Birke & Holger Dette, 2007. "Estimating a Convex Function in Nonparametric Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(2), pages 384-404, June.
    5. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    6. Suoniemi, Ilpo, 1994. "Non-Parametric Estimation of Lorenz Curves Using Locally Weighted Regression," Discussion Papers 59, VATT Institute for Economic Research.
    7. Darinka Dentcheva & Andrzej Ruszczynski, 2005. "Inverse stochastic dominance constraints and rank dependent expected utility theory," GE, Growth, Math methods 0503001, University Library of Munich, Germany.
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    Cited by:

    1. Darinka Dentcheva & Spiridon Penev & Andrzej Ruszczyński, 2010. "Kusuoka representation of higher order dual risk measures," Annals of Operations Research, Springer, vol. 181(1), pages 325-335, December.
    2. Dentcheva Darinka & Stock Gregory J. & Rekeda Ludmyla, 2011. "Mean-risk tests of stochastic dominance," Statistics & Risk Modeling, De Gruyter, vol. 28(2), pages 97-118, May.
    3. Darinka Dentcheva & Spiridon Penev & Andrzej Ruszczyński, 2017. "Statistical estimation of composite risk functionals and risk optimization problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(4), pages 737-760, August.

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