Skip to main content
Log in

Calibration of estimator-weights via semismooth Newton method

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

Weighting is a common methodology in survey statistics to increase accuracy of estimates or to compensate for non-response. One standard approach for weighting is calibration estimation which represents a common numerical problem. There are various approaches in the literature available, but quite a number of distance-based approaches lack a mathematical justification or are numerically unstable. In this paper we reformulate the calibration problem as a system of nonlinear equations. Although the equations are lacking differentiability properties, one can show that they are semismooth and the corresponding extension of Newton’s method is applicable. This is a mathematically rigorous approach and the numerical results show the applicability of this method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Clarke F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    Google Scholar 

  2. Demnati A., Rao J.: Linearization variance estimators for survey data. Surv. Methodol. 30, 17–26 (2004)

    Google Scholar 

  3. Dennis J., Schnabel R.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs NJ (1983)

    Google Scholar 

  4. Deville J.C., Särndal C.E.: Calibration estimators in survey sampling. J. Am. Stat. Assoc. 87, 376–382 (1992)

    Article  Google Scholar 

  5. Deville J.C., Särndal C.E., Sautory O.: Generalized raking procedures in survey sampling. J. Am. Stat. Assoc. 88, 1013–1020 (1993)

    Article  Google Scholar 

  6. Estevao V., Särndal C.E.: Survey estimates by calibration on complex auxiliary information. Int. Stat. Rev. 74, 127–147 (2006)

    Article  Google Scholar 

  7. Fischer A.: Solution of monotone complementarity problems with locally lipschitzian functions. Math. Program. 76, 513–532 (1997)

    Google Scholar 

  8. Gill P., Murray W., Wright M.: Practical Optimization. Academic Press, NY (1981)

    Google Scholar 

  9. Horst R.: Nichtlineare Optimierung. Carl Hanser Verlag, USA (1979)

    Google Scholar 

  10. Horvitz D., Thompson D.: A generalization of sampling without replacement from a finite universe. J. Am. Stat. Assoc. 47, 663–685 (1952)

    Article  Google Scholar 

  11. Kim J., Park M.: Calibration estimation in survey sampling. Int. Stat. Rev. 78, 21–39 (2011)

    Article  Google Scholar 

  12. Kott P.: Using calibration weighting to adjust for nonresponse and coverage errors. Surv. Methodol. 32, 133–142 (2006)

    Google Scholar 

  13. Mifflin R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Contr. Optim. 15, 957–972 (1977)

    Article  Google Scholar 

  14. Münnich, R., Sachs, E., Wagner, M.: Numerical solution of optimal allocation problems in stratified sampling under box constraints. (In submission) (2011)

  15. Pang J.S., Qi L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993)

    Article  Google Scholar 

  16. Qi L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    Article  Google Scholar 

  17. Qi L., Sun J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)

    Article  Google Scholar 

  18. R Development Core Team: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2010). http://www.R-project.org/. ISBN 3-900051-07-0

  19. Rademacher H.: über partielle und totale differenzierbarkeit von funktionen mehrerer variablen und über die transformation der doppelintegrale. Mathematische Annalen 79, 340–359 (1919)

    Article  Google Scholar 

  20. Ruszczynski A.: Nonlinear Optimization. Princeton University Press, Princeton (2006)

    Google Scholar 

  21. Särndal C.E.: The calibration approach in survey theory and practise. Surv. Methodol. 33, 99–119 (2007)

    Google Scholar 

  22. Shapiro A.: On concepts of directional differentiability. J. Optim. Theory Appl. 66, 477–487 (1990)

    Article  Google Scholar 

  23. Singh A., Mohl C.: Understanding calibration estimators in survey sampling. Surv. Methodol. 22, 107–115 (1996)

    Google Scholar 

  24. Stukel D., Hidiroglou M., Särndal C.E.: Variance estimation for calibration estimators: a comparison of jackknifing versus taylor linearization. Surv. Methodol. 22, 117–125 (1996)

    Google Scholar 

  25. Tillé, Y., Matei, A.: sampling: survey sampling (2009). http://CRAN.R-project.org/package=sampling. R package version 2.3

  26. Vanderhoeft, C.: Statistics belgium working papers—generalised calibration at statistics belgium. Tech. Rep. STATBEL, 1–192 (2001)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Matthias Wagner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Münnich, R.T., Sachs, E.W. & Wagner, M. Calibration of estimator-weights via semismooth Newton method. J Glob Optim 52, 471–485 (2012). https://doi.org/10.1007/s10898-011-9759-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-011-9759-1

Keywords

Navigation