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.
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References
Clarke F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Demnati A., Rao J.: Linearization variance estimators for survey data. Surv. Methodol. 30, 17–26 (2004)
Dennis J., Schnabel R.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs NJ (1983)
Deville J.C., Särndal C.E.: Calibration estimators in survey sampling. J. Am. Stat. Assoc. 87, 376–382 (1992)
Deville J.C., Särndal C.E., Sautory O.: Generalized raking procedures in survey sampling. J. Am. Stat. Assoc. 88, 1013–1020 (1993)
Estevao V., Särndal C.E.: Survey estimates by calibration on complex auxiliary information. Int. Stat. Rev. 74, 127–147 (2006)
Fischer A.: Solution of monotone complementarity problems with locally lipschitzian functions. Math. Program. 76, 513–532 (1997)
Gill P., Murray W., Wright M.: Practical Optimization. Academic Press, NY (1981)
Horst R.: Nichtlineare Optimierung. Carl Hanser Verlag, USA (1979)
Horvitz D., Thompson D.: A generalization of sampling without replacement from a finite universe. J. Am. Stat. Assoc. 47, 663–685 (1952)
Kim J., Park M.: Calibration estimation in survey sampling. Int. Stat. Rev. 78, 21–39 (2011)
Kott P.: Using calibration weighting to adjust for nonresponse and coverage errors. Surv. Methodol. 32, 133–142 (2006)
Mifflin R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Contr. Optim. 15, 957–972 (1977)
Münnich, R., Sachs, E., Wagner, M.: Numerical solution of optimal allocation problems in stratified sampling under box constraints. (In submission) (2011)
Pang J.S., Qi L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993)
Qi L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)
Qi L., Sun J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)
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
Rademacher H.: über partielle und totale differenzierbarkeit von funktionen mehrerer variablen und über die transformation der doppelintegrale. Mathematische Annalen 79, 340–359 (1919)
Ruszczynski A.: Nonlinear Optimization. Princeton University Press, Princeton (2006)
Särndal C.E.: The calibration approach in survey theory and practise. Surv. Methodol. 33, 99–119 (2007)
Shapiro A.: On concepts of directional differentiability. J. Optim. Theory Appl. 66, 477–487 (1990)
Singh A., Mohl C.: Understanding calibration estimators in survey sampling. Surv. Methodol. 22, 107–115 (1996)
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)
Tillé, Y., Matei, A.: sampling: survey sampling (2009). http://CRAN.R-project.org/package=sampling. R package version 2.3
Vanderhoeft, C.: Statistics belgium working papers—generalised calibration at statistics belgium. Tech. Rep. STATBEL, 1–192 (2001)
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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
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DOI: https://doi.org/10.1007/s10898-011-9759-1