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On the estimation of the general parameter

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  • Stearns, Matthew
  • Singh, Sarjinder

Abstract

In this paper, we first discuss the origin, developments and various thoughts by several researchers on the generalized linear regression estimator (GREG) due to Deville and Särndal [Deville, J.C., Särndal, C.E., 1992. Calibration estimators in survey sampling. J. Amer. Statist. Assoc. 87, 376-382]. Then, the problem of estimation of the general parameter of interest considered by Rao [Rao, J.N.K., 1994. Estimating totals and distribution functions using auxiliary information at the estimation stage. J. Official Statist. 10 (2), 153-165], and Singh [Singh, S., 2001. Generalized calibration approach for estimating the variance in survey sampling. Ann. Inst. Statist. Math. 53 (2), 404-417; Singh, S., 2004. Golden and Silver Jubilee Year-2003 of the linear regression estimators. In: Proceedings of the Joint Statistical Meeting, Toronto (Available on the CD), 4382-4380; Singh, S., 2006. Survey statisticians celebrate Golden Jubilee Year-2003 of the linear regression estimator. Metrika 1-18] is further investigated. In addition to that it is shown that the Farrell and Singh [Farrell, P.J., Singh, S., 2005. Model-assisted higher order calibration of estimators of variance. Australian & New Zealand J. Statist. 47 (3), 375-383] estimators are also a special case of the proposed methodology. Interestingly, it has been noted that the single model assisted calibration constraint studied by Farrell and Singh [Farrell, P.J., Singh, S., 2002. Re-calibration of higher order calibration weights. Presented at Statistical Society of Canada conference, Hamilton (Available on CD); Farrell, P.J., Singh, S., 2005. Model-assisted higher order calibration of estimators of variance. Australian & New Zealand J. Statist. 47 (3), 375-383] and Wu [Wu, C., 2003. Optimal calibration estimators in survey sampling. Biometrika 90, 937-951] is not helpful for calibrating the Sen [Sen, A.R., 1953. On the estimate of the variance in sampling with varying probabilities. J. Indian Soc. Agril. Statist. 5, 119-127] and Yates and Grundy [Yates, F., Grundy, P.M., 1953. Selection without replacement from within strata with probability proportional to size. J. Roy. Statist. Soc. Ser. 15, 253-261] estimator of the variance of the linear regression estimator under the optimal designs of Godambe and Joshi [Godambe, V.P., Joshi, V.M., 1965. Admissibility and Bayes estimation in sampling finite populations--I. Ann. Math. Statist. 36, 1707-1722]. Three new estimators of the variance of the proposed linear regression type estimator of the general parameters of interest are introduced and compared with each other. The newly proposed two-dimensional linear regression models are found to be useful, unlike a simulation based on a couple of thousands of random samples, in comparing the estimators of variance. The use of knowledge of the model parameters in assisting the estimators of variance has been found to be beneficial. The most attractive feature is that it has been shown theoretically that the proposed method of calibration always remains more efficient than the GREG estimator.

Suggested Citation

  • Stearns, Matthew & Singh, Sarjinder, 2008. "On the estimation of the general parameter," Computational Statistics & Data Analysis, Elsevier, vol. 52(9), pages 4253-4271, May.
    • Handle: RePEc:eee:csdana:v:52:y:2008:i:9:p:4253-4271
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    References listed on IDEAS

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    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.
    2. Wu C. & Sitter R. R, 2001. "A Model-Calibration Approach to Using Complete Auxiliary Information From Survey Data," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 185-193, March.
    3. Rueda, M. & Martinez, S. & Martinez, H. & Arcos, A., 2006. "Mean estimation with calibration techniques in presence of missing data," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3263-3277, July.
    4. Sarjinder Singh, 2006. "Survey statisticians celebrate golden jubilee year 2003 of the linear regression estimator," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 63(1), pages 1-18, February.
    5. Changbao Wu, 2003. "Optimal calibration estimators in survey sampling," Biometrika, Biometrika Trust, vol. 90(4), pages 937-951, December.
    6. Sitter R.R. & Wu C., 2002. "Efficient Estimation of Quadratic Finite Population Functions in the Presence of Auxiliary Information," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 535-543, June.
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    Cited by:

    1. Sarjinder Singh, 2012. "On the calibration of design weights using a displacement function," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(1), pages 85-107, January.
    2. Ohyama, Tetsuji, 2013. "Prior value incorporated calibration estimator in stratified random sampling," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 46-51.
    3. J. Al-Jararha & Mazen Sulaiman, 2020. "Horvitz-Thompson estimator based on the auxiliary variable," Statistics in Transition New Series, Polish Statistical Association, vol. 21(1), pages 37-53, March.
    4. Martínez, S. & Rueda, M. & Arcos, A. & Martínez, H. & Sánchez-Borrego, I., 2011. "Post-stratified calibration method for estimating quantiles," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 838-851, January.
    5. Al-Jararha J. & Sulaiman Mazen, 2020. "Horvitz-Thompson estimator based on theauxiliary variable," Statistics in Transition New Series, Polish Statistical Association, vol. 21(1), pages 37-54, March.

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