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A Semi-Parametric Basis for Combining Estimation Problems Under Quadratic Loss

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  • Judge, George G.
  • Mittelhammer, Ron C

Abstract

When there is uncertainty concerning the appropriate statistical model to use in representing the data sampling process and corresponding estimators, we consider a basis for optimally combining estimation problems. In the context of the multivariate linear statistical model, we consider a semi-parametric Stein-like (SPSL) estimator, B( ), that shrinks to a random data-dependent vector and, under quadratic loss, has superior performance relative to the conventional least squares estimator. The relationship of the SPSL estimator to the family of Stein estimators is noted and risk dominance extensions between correlated estimators are demonstrated. As an application we consider the problem of a possibly ill-conditioned design matrix and devise a corresponding SPSL estimator. Asymptotic and analytic finite sample risk properties of the estimator are demonstrated. An extensive sampling experiment is used to investigate finite sample performance over a wide range of data sampling processes to illustrate the robustness of the estimator for an array of symmetric and skewed distributions. Bootstrapping procedures are used to develop confidence sets and a basis for inference.

Suggested Citation

  • Judge, George G. & Mittelhammer, Ron C, 2003. "A Semi-Parametric Basis for Combining Estimation Problems Under Quadratic Loss," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt8z25j0w3, Department of Agricultural & Resource Economics, UC Berkeley.
    • Handle: RePEc:cdl:agrebk:qt8z25j0w3
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    References listed on IDEAS

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    1. Ullah, Aman & Ullah, Shobha, 1978. "Double k-Class Estimators of Coefficients in Linear Regression," Econometrica, Econometric Society, vol. 46(3), pages 705-722, May.
    2. Kim, Tae-Hwan & White, Halbert, 2000. "James-Stein Type Estimators in Large Samples with Application to the Least Absolute Deviations Estimator," University of California at San Diego, Economics Working Paper Series qt4zq9k3qh, Department of Economics, UC San Diego.
    3. Kim, Tae-Hwan & White, Halbert, 1999. "James-Stein Type Estimators in Large Samples with Application to the Least Absolute Deviations Estimator," University of California at San Diego, Economics Working Paper Series qt9914w10r, Department of Economics, UC San Diego.
    4. Kim, Tae-Hwan & White, Halbert, 2000. "James-Stein Type Estimator in Large Samples with Application to the Least Absolute Deviations Estimator," University of California at San Diego, Economics Working Paper Series qt3mn102zs, Department of Economics, UC San Diego.
    5. Kim T-H. & White H., 2001. "James-Stein-Type Estimators in Large Samples With Application to the Least Absolute Deviations Estimator," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 697-705, June.
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    Citations

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    Cited by:

    1. Marian Grendar & George Judge, 2008. "Large-Deviations Theory and Empirical Estimator Choice," Econometric Reviews, Taylor & Francis Journals, vol. 27(4-6), pages 513-525.
    2. Judge, George G. & Mittelhammer, Ron C, 2004. "Estimating the Link Function in Multinomial Response Models under Endogeneity and Quadratic Loss," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt4422n50w, Department of Agricultural & Resource Economics, UC Berkeley.
    3. Miller, Douglas J. & Mittelhammer, Ronald C. & Judge, George G., 2004. "Entropy-Based Estimation And Inference In Binary Response Models Under Endogeneity," 2004 Annual meeting, August 1-4, Denver, CO 20319, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association).
    4. Tae-Hwan Kim & Christophe Muller, 2020. "Inconsistency transmission and variance reduction in two-stage quantile regression," Post-Print hal-02084505, HAL.
    5. Shakhawat Hossain & Shahedul A. Khan, 2020. "Shrinkage estimation of the exponentiated Weibull regression model for time‐to‐event data," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 74(4), pages 592-610, November.
    6. Raheem, S.M. Enayetur & Ahmed, S. Ejaz & Doksum, Kjell A., 2012. "Absolute penalty and shrinkage estimation in partially linear models," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 874-891.
    7. Cl'ement de Chaisemartin & Xavier D'Haultf{oe}uille, 2020. "Empirical MSE Minimization to Estimate a Scalar Parameter," Papers 2006.14667, arXiv.org.
    8. Mittelhammer, Ron C. & Judge, George G., 2005. "Combining estimators to improve structural model estimation and inference under quadratic loss," Journal of Econometrics, Elsevier, vol. 128(1), pages 1-29, September.
    9. Mittelhammer, Ron C & Judge, George G. & Miller, Douglas J & Cardell, N. Scott, 2005. "Minimum Divergence Moment Based Binary Response Models: Estimation and Inference," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt1546s6rn, Department of Agricultural & Resource Economics, UC Berkeley.
    10. Joshua D. Angrist & Peter D. Hull & Parag A. Pathak & Christopher R. Walters, 2017. "Leveraging Lotteries for School Value-Added: Testing and Estimation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 132(2), pages 871-919.
    11. Mittelhammer, Ron C & Judge, George G. & Miller, Douglas J & Cardell, N. Scott, 2005. "Minimum Divergence Moment Based Binary Response Models: Estimation and Inference," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt1546s6rn, Department of Agricultural & Resource Economics, UC Berkeley.
    12. Ruoyao Shi, 2021. "An Averaging Estimator for Two Step M Estimation in Semiparametric Models," Working Papers 202105, University of California at Riverside, Department of Economics.
    13. Shakhawat Hossain & Trevor Thomson & Ejaz Ahmed, 2018. "Shrinkage estimation in linear mixed models for longitudinal data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(5), pages 569-586, July.
    14. Judge, George G. & Mittelhammer, Ron C, 2004. "Estimating the Link Function in Multinomial Response Models under Endogeneity and Quadratic Loss," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt4422n50w, Department of Agricultural & Resource Economics, UC Berkeley.
    15. Judge, George G. & Mittelhammer, Ron C., 2007. "Estimation and inference in the case of competing sets of estimating equations," Journal of Econometrics, Elsevier, vol. 138(2), pages 513-531, June.
    16. Marchand, Éric & Strawderman, William E., 2020. "On shrinkage estimation for balanced loss functions," Journal of Multivariate Analysis, Elsevier, vol. 175(C).

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