Review Articles

Adjusted variance estimators based on minimizing mean squared error for stratified random samples

Guoyi Zhang ,

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, USA

gzhang123@gmail.com

Bruce Swan

Department of Mathematics, SUNY Buffalo State College, Buffalo, NY, USA

Pages | Received 31 May. 2023, Accepted 03 Jan. 2024, Published online: 16 Jan. 2024,
  • Abstract
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In the realm of survey data analysis, encountering substantial variance relative to bias is a common occurrence. In this study, we present an innovative strategy to tackle this issue by introducing slightly biased variance estimators. These estimators incorporate a constant c within the range of 0 to 1, which is determined through the minimization of Mean Squared Error (MSE) for c×(variance estimator). This research builds upon the foundation laid by Kourouklis (2012, A new estimator of the variance based on minimizing mean squared error. The American Statistician66(4), 234–236) and extends their work into the domain of survey sampling. Extensive simulation studies are conducted to illustrate the superior performance of the adjusted variance estimators when compared to standard variance estimators, particularly in terms of MSE. These findings underscore the efficacy of our proposed approach in enhancing the precision of variance estimation within the context of survey data analysis.

References

  • Brewster, J. F., & Zidek, J. V. (1974). Improving on equivariant estimators. The Annals of Statistics2(1), 21–38.
  • Brown, L. D. (1968). Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters. Annals of Mathematical Statistics39(1), 29–48. https://doi.org/10.1214/aoms/1177698503
  • Espejo, M., Pineda, M., & Nadarajah, S. (2013). Optimal unbiased estimation of some population central moments. Metron71(1), 39–62. https://doi.org/10.1007/s40300-013-0006-z
  • Heffernan, P. (1997). Unbiased estimation of central moments by using u -statistics. Journal of the Royal Statistical Society Series B59(4), 861–863. https://doi.org/10.1111/1467-9868.00102
  • Kourouklis, S. (2012). A new estimator of the variance based on minimizing mean squared error. The American Statistician66(4), 234–236. https://doi.org/10.1080/00031305.2012.735209
  • Lohr, S. (2010). Sampling: Design and analysis (2nd ed.). Cengage Learning.
  • Maruyama, Y. (1998). Minimax estimators of a normal variance. Metrika48(3), 209–214. https://doi.org/10.1007/PL00003974
  • Maruyama, Y., & Strawderman, W. E. (2006). A new class of minimax generalized Bayes estimators of a normal variance. Journal of Statistical Planning and Inference136(11), 3822–3836. https://doi.org/10.1016/j.jspi.2005.05.005
  • Stein, C. (1964). Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean. Annals of the Institute of Statistical Mathematics16(1), 155–160. https://doi.org/10.1007/BF02868569
  • Strawderman, W. E. (1974). Minimax estimation of powers of the variance of a normal population under squared error loss. The Annals of Statistics2(1), 190–198. https://doi.org/10.1214/aos/1176342625
  • Sukhatme, P. V. (1984). Sampling theory of surveys with applications. Iowa State College Press.
  • Yatracos, Y. (2005). Artificially augmented samples, shrinkage, and mean squared error reduction. Journal of the American Statistical Association100(472), 1168–1175. https://doi.org/10.1198/016214505000000321

To cite this article: Guoyi Zhang & Bruce Swan (2024) Adjusted variance estimators based on minimizing mean squared error for stratified random samples, Statistical Theory and Related Fields, 8:2, 117-123, DOI: 10.1080/24754269.2024.2303915

To link to this article: https://doi.org/10.1080/24754269.2024.2303915