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Efficient estimation of linear functionals of a bivariate distribution with equal, but unknown marginals: the least-squares approach

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  • Peng, Hanxiang
  • Schick, Anton

Abstract

In this paper, we characterize and construct efficient estimators of linear functionals of a bivariate distribution with equal marginals. An efficient estimator equals the empirical estimator minus a correction term and provides significant improvements over the empirical estimator. We construct an efficient estimator by estimating the correction term. For this we use the least-squares principle and an estimated orthonormal basis for the Hilbert space of square-integrable functions under the unknown equal marginal distribution. Simulations confirm the asymptotic behavior of this estimator in moderate sample sizes and the considerable theoretical gains over the empirical estimator.

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  • Peng, Hanxiang & Schick, Anton, 2005. "Efficient estimation of linear functionals of a bivariate distribution with equal, but unknown marginals: the least-squares approach," Journal of Multivariate Analysis, Elsevier, vol. 95(2), pages 385-409, August.
    • Handle: RePEc:eee:jmvana:v:95:y:2005:i:2:p:385-409
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    References listed on IDEAS

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    1. Forrester Jeffrey S. & Hooper William J. & Peng Hanxiang & Schick Anton, 2003. "On the construction of efficient estimators in semiparametric models," Statistics & Risk Modeling, De Gruyter, vol. 21(2/2003), pages 109-138, February.
    2. Newey, Whitney K., 1997. "Convergence rates and asymptotic normality for series estimators," Journal of Econometrics, Elsevier, vol. 79(1), pages 147-168, July.
    3. Modarres, Reza, 2003. "Estimation of a bivariate symmetric distribution function," Statistics & Probability Letters, Elsevier, vol. 63(1), pages 25-34, May.
    4. Peng, Hanxiang & Schick, Anton, 2002. "On efficient estimation of linear functionals of a bivariate distribution with known marginals," Statistics & Probability Letters, Elsevier, vol. 59(1), pages 83-91, August.
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    Cited by:

    1. Ursula U. Müller & Hanxiang Peng & Anton Schick, 2019. "Inference about the slope in linear regression: an empirical likelihood approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(1), pages 181-211, February.
    2. Segers, J.J.J. & van den Akker, R. & Werker, B.J.M., 2008. "Improving Upon the Marginal Empirical Distribution Functions when the Copula is Known," Discussion Paper 2008-40, Tilburg University, Center for Economic Research.
    3. Peng, Hanxiang & Schick, Anton, 2018. "Asymptotic normality of quadratic forms with random vectors of increasing dimension," Journal of Multivariate Analysis, Elsevier, vol. 164(C), pages 22-39.
    4. Segers, J.J.J. & van den Akker, R. & Werker, B.J.M., 2008. "Improving Upon the Marginal Empirical Distribution Functions when the Copula is Known," Other publications TiSEM 950a8cda-8f8c-43a9-a5c2-8, Tilburg University, School of Economics and Management.
    5. Peng Hanxiang & Schick Anton, 2004. "Efficient estimation of a linear functional of a bivariate distribution with equal, but unknown, marginals: The minimum chi-square approach," Statistics & Risk Modeling, De Gruyter, vol. 22(4/2004), pages 301-318, April.

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