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Estimation of the Boundary of a Variable Observed with A Symmetric Error

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  • Florens, Jean-Pierre
  • Simar, Léopold
  • Van Keilegom, Ingrid

Abstract

Consider the model with , where tau is an unknown constant (the boundary of X), Z is a random variable defined on , epsilon is a symmetric error, and epsilon and Z are independent. Based on an iid sample of Y, we aim at identifying and estimating the boundary tau when the law of epsilon is unknown (apart from symmetry) and in particular its variance is unknown. We propose an estimation procedure based on a minimal distance approach and by making use of Laguerre polynomials. Asymptotic results as well as finite sample simulations are shown. The paper also proposes an extension to stochastic frontier analysis, where the model is conditional to observed variables. The model becomes , where Y is a cost, w(1) are the observed outputs and w(2) represents the observed values of other conditioning variables, so Z is the cost inefficiency. Some simulations illustrate again how the approach works in finite samples, and the proposed procedure is illustrated with data coming from post offices in France.
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Suggested Citation

  • Florens, Jean-Pierre & Simar, Léopold & Van Keilegom, Ingrid, 2019. "Estimation of the Boundary of a Variable Observed with A Symmetric Error," TSE Working Papers 19-990, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:33355
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    References listed on IDEAS

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    1. Kneip, Alois & Simar, Léopold & Van Keilegom, Ingrid, 2015. "Frontier estimation in the presence of measurement error with unknown variance," Journal of Econometrics, Elsevier, vol. 184(2), pages 379-393.
    2. Cristina Butucea & Pierre Vandekerkhove, 2014. "Semiparametric Mixtures of Symmetric Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 227-239, March.
    3. Lucie Aarts & Piet Groeneboom & Geurt Jongbloed, 2007. "Estimating the Upper Support Point in Deconvolution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(3), pages 552-568, September.
    4. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2020. "Robust frontier estimation from noisy data: A Tikhonov regularization approach," Econometrics and Statistics, Elsevier, vol. 14(C), pages 1-23.
    5. Schwarz, Maik & Van Bellegem, Sébastien, 2010. "Consistent density deconvolution under partially known error distribution," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 236-241, February.
    6. Fan, Yanqin & Li, Qi & Weersink, Alfons, 1996. "Semiparametric Estimation of Stochastic Production Frontier Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(4), pages 460-468, October.
    7. Cazals, Catherine & Florens, Jean-Pierre & Simar, Leopold, 2002. "Nonparametric frontier estimation: a robust approach," Journal of Econometrics, Elsevier, vol. 106(1), pages 1-25, January.
    8. Cazals, Catherine & Fève, Frédérique & Florens, Jean-Pierre & Simar, Léopold, 2016. "Nonparametric instrumental variables estimation for efficiency frontier," Journal of Econometrics, Elsevier, vol. 190(2), pages 349-359.
    9. Aurore Delaigle & Peter Hall, 2016. "Methodology for non-parametric deconvolution when the error distribution is unknown," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 231-252, January.
    10. Goldenshluger, A. & Tsybakov, A., 2004. "Estimating the endpoint of a distribution in the presence of additive observation errors," Statistics & Probability Letters, Elsevier, vol. 68(1), pages 39-49, June.
    11. Delaigle, A. & Gijbels, I., 2006. "Data-driven boundary estimation in deconvolution problems," Computational Statistics & Data Analysis, Elsevier, vol. 50(8), pages 1965-1994, April.
    12. Qi Li & Juan Lin & Jeffrey S. Racine, 2013. "Optimal Bandwidth Selection for Nonparametric Conditional Distribution and Quantile Functions," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(1), pages 57-65, January.
    13. Bertrand, Aurelie & Van Keilegom, Ingrid & Legrand, Catherine, 2017. "Flexible parametric approach to classical measurement error variance estimation without auxiliary data," LIDAM Discussion Papers ISBA 2017025, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    14. Schwarz, M. & Van Bellegem, S., 2010. "Consistent density deconvolution under partially known error distribution," LIDAM Reprints ISBA 2010013, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    15. Daouia, Abdelaati & Simar, Leopold, 2007. "Nonparametric efficiency analysis: A multivariate conditional quantile approach," Journal of Econometrics, Elsevier, vol. 140(2), pages 375-400, October.
    16. Timo Kuosmanen & Mika Kortelainen, 2012. "Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints," Journal of Productivity Analysis, Springer, vol. 38(1), pages 11-28, August.
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    Cited by:

    1. Oleg Badunenko & Daniel J. Henderson, 2024. "Production analysis with asymmetric noise," Journal of Productivity Analysis, Springer, vol. 61(1), pages 1-18, February.
    2. Caitlin O’Loughlin & Léopold Simar & Paul W. Wilson, 2023. "Methodologies for assessing government efficiency," Chapters, in: António Afonso & João Tovar Jalles & Ana Venâncio (ed.), Handbook on Public Sector Efficiency, chapter 4, pages 72-101, Edward Elgar Publishing.
    3. Centorrino, Samuele & Parmeter, Christopher F., 2024. "Nonparametric estimation of stochastic frontier models with weak separability," Journal of Econometrics, Elsevier, vol. 238(2).
    4. Léopold Simar & Paul W. Wilson, 2023. "Nonparametric, Stochastic Frontier Models with Multiple Inputs and Outputs," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 41(4), pages 1391-1403, October.
    5. William C. Horrace & Yulong Wang, 2022. "Nonparametric tests of tail behavior in stochastic frontier models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 37(3), pages 537-562, April.
    6. Kamil Makieła & Błażej Mazur, 2022. "Model uncertainty and efficiency measurement in stochastic frontier analysis with generalized errors," Journal of Productivity Analysis, Springer, vol. 58(1), pages 35-54, August.
    7. Kamil Makieła & Błażej Mazur, 2020. "Bayesian Model Averaging and Prior Sensitivity in Stochastic Frontier Analysis," Econometrics, MDPI, vol. 8(2), pages 1-22, April.
    8. Kamil Makie{l}a & B{l}a.zej Mazur, 2020. "Stochastic Frontier Analysis with Generalized Errors: inference, model comparison and averaging," Papers 2003.07150, arXiv.org, revised Oct 2020.
    9. Jun Cai & William C. Horrace & Christopher F. Parmeter, 2024. "Penalized sieve estimation of zero‐inefficiency stochastic frontiers," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 39(1), pages 41-65, January.

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    Keywords

    Characteristic function; cumulant function; flexible parametric family; frontier estimation; Laguerre polynomials;
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