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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.

Suggested Citation

  • 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. Modarres, Reza, 2003. "Estimation of a bivariate symmetric distribution function," Statistics & Probability Letters, Elsevier, vol. 63(1), pages 25-34, May.
    2. 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.
    3. 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.
    4. Newey, Whitney K., 1997. "Convergence rates and asymptotic normality for series estimators," Journal of Econometrics, Elsevier, vol. 79(1), pages 147-168, July.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    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. 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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