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Optimal unbiased estimation of some population central moments


  • Mariano Ruiz Espejo
  • Miguel Delgado Pineda
  • Saralees Nadarajah



In this paper, we have treated the problem of estimating some population central moments under distribution-free setting. Uniformly minimum variance unbiased estimators for some population central moments have been derived. Some examples of unbiased estimators of central moments have been given under various sampling designs such as simple random sampling with replacement (srsr) or without replacement (srs), probability proportional to size with replacement (ppsr) and probability graduated variable proportional to size without replacement (pgvps). An optimal unbiased estimator of the third population central moment is proposed and extended to some real situations. Some optimal unbiased estimators of the fourth population central moment are given. Several optimal unbiased estimators of the variance of the “sample quasivariance estimator” are identified. Finally, computer programs in R implementing all of the estimators are given. Copyright Sapienza Università di Roma 2013

Suggested Citation

  • Mariano Ruiz Espejo & Miguel Delgado Pineda & Saralees Nadarajah, 2013. "Optimal unbiased estimation of some population central moments," METRON, Springer;Sapienza Università di Roma, vol. 71(1), pages 39-62, June.
  • Handle: RePEc:spr:metron:v:71:y:2013:i:1:p:39-62
    DOI: 10.1007/s40300-013-0006-z

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    References listed on IDEAS

    1. Sarjinder Singh, 2001. "Generalized Calibration Approach for Estimating Variance in Survey Sampling," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(2), pages 404-417, June.
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    4. Ping Wu & Li Xing Zhu, 2010. "An Orthogonality-Based Estimation of Moments for Linear Mixed Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(2), pages 253-263.
    5. Angela Montanari & Cinzia Viroli, 2010. "A skew-normal factor model for the analysis of student satisfaction towards university courses," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(3), pages 473-487.
    6. Guiso, Luigi & Jappelli, Tullio & Pistaferri, Luigi, 2002. "An Empirical Analysis of Earnings and Employment Risk," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(2), pages 241-253, April.
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