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Estimating a Convex Function in Nonparametric Regression

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  • MELANIE BIRKE
  • HOLGER DETTE
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    Abstract

    A new nonparametric estimate of a convex regression function is proposed and its stochastic properties are studied. The method starts with an unconstrained estimate of the derivative of the regression function, which is firstly isotonized and then integrated. We prove asymptotic normality of the new estimate and show that it is first order asymptotically equivalent to the initial unconstrained estimate if the regression function is in fact convex. If convexity is not present, the method estimates a convex function whose derivative has the same "L"-super-"p"-norm as the derivative of the (non-convex) underlying regression function. The finite sample properties of the new estimate are investigated by means of a simulation study and it is compared with a least squares approach of convex estimation. The application of the new method is demonstrated in two data examples. Copyright 2007 Board of the Foundation of the Scandinavian Journal of Statistics..

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    Bibliographic Info

    Article provided by Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association in its journal Scandinavian Journal of Statistics.

    Volume (Year): 34 (2007)
    Issue (Month): 2 ()
    Pages: 384-404

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    Handle: RePEc:bla:scjsta:v:34:y:2007:i:2:p:384-404

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    Cited by:
    1. 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.
    2. Richard Blundell & Dennis Kristensen & Rosa Matzkin, 2011. "Bounding quantile demand functions using revealed preference inequalities," CeMMAP working papers CWP21/11, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Leorato, S., 2009. "A refined Jensen's inequality in Hilbert spaces and empirical approximations," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 1044-1060, May.

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