Article |
Use of functionals in linearization and composite estimation with application to two-sample survey data
Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 Avenue Alain Savary, 21078 Dijon, France camelia.goga{at}u-bourgogne.fr
Laboratoire de Statistique d'Enquête, ENSAI/CREST, rue Blaise Pascal, Campus de Ker Lann, 35170 Bruz, France deville{at}ensai.fr
Toulouse School of Economics, Université Toulouse 1, 21 allée de Brienne, 31000 Toulouse, France ruiz{at}cict.fr
Received for publication 1 July 2006. Revision received 1 December 2008.
An important problem associated with two-sample surveys is the estimation of nonlinear functions of finite population totals such as ratios, correlation coefficients or measures of income inequality. Computation and estimation of the variance of such complex statistics are made more difficult by the existence of overlapping units. In one-sample surveys, the linearization method based on the influence function approach is a powerful tool for variance estimation. We introduce a two-sample linearization technique that can be viewed as a generalization of the one-sample influence function approach. Our technique is based on expressing the parameters of interest as multivariate functionals of finite and discrete measures and then using partial influence functions to compute the linearized variables. Under broad assumptions, the asymptotic variance of the substitution estimator, derived from Deville (1999), is shown to be the variance of a weighted sum of the linearized variables. The paper then focuses on a general class of composite substitution estimators, and from this class the optimal estimator for minimizing the asymptotic variance is obtained. The efficiency of the optimal composite estimator is demonstrated through an empirical study.
Key Words: Gini index change • Partial influence function • Substitution estimator • Two-dimensional sampling design • Variance estimation • Variance optimization
References
-
Bell P. Comparison of alternative labour force survey estimators. Survey Methodol. (2001) 27:53–64.
Berger Y. Variance estimation for measures of change in probability sampling. Can. J. Statist. (2004) 32:451–67.[CrossRef]
Berger Y., Skinner C. A jackknife variance estimator for unequal probability sampling. J. R. Statist. Soc. (2005) B 67:79–89.[CrossRef]
Breidt F. J., Opsomer J. D. Local polynomial regression estimators in survey sampling. Ann. Statist. (2000) 28:1026–53.[CrossRef]
Campbell C. A different view of finite population estimation. In: Proc. Survey Res. Meth. Sect. Am. Statist. Assoc. (1980) Washington, DC. 319–24.
Canty A. J., Davison A. C. Resampling-based variance estimation for labour force surveys. Statistician (1999) 48:379–91.
Cochran W. G. Sampling Techniques (1977) New York: Wiley.
Demnati A., Rao J. Linearization variance estimators for survey data. Survey Methodol. (2004) 30:17–26.
Deville J. Variance estimation for complex statistics and estimators: linearization and residual techniques. Survey Methodol. (1999) 25:193–203.
Eckler A. R. Rotation sampling. Ann. Math. Statist. (1955) 26:664–85.[CrossRef]
Fernholz L. T. Von Mises Calculus for Statistical Functionals (1983) New York: Springer. Lecture Notes in Statistics 19.
Fuller W. Regression estimation for survey samples. Survey Methodol. (2002) 28:5–23.
Fuller W., Rao J. A regression composite estimator with application to the Canadian labour force survey. Survey Methodol. (2001) 27:45–52.
Hampel F. R. The influence curve and its role in robust statistics. J. Am. Statist. Assoc. (1974) 69:383–93.[CrossRef][Web of Science]
Hidiroglou M. A. Double sampling. Survey Methodol. (2001) 27:143–54.
Huber P. J. Robust Statistics (1981) New York: Wiley.
Isaki C. T., Fuller W. A. Survey design under the regression superpopulation model. J. Am. Statist. Assoc. (1982) 77:89–96.[CrossRef][Web of Science]
Jessen R. J. Statistical investigation of a farm survey for obtaining farm facts. Iowa Agric. Station Res. Bull. (1942) 304:54–59.
Kish L. Survey Sampling (1965) New York: Wiley.
Kovacevi
M., Binder D. A. Variance estimation for measures of income inequality and polarization—the estimating equations approach. J. Offic. Statist. (1997) 13:41–58.
Merkouris T. Cross-sectional estimation in multiple-panel household surveys. Survey Methodol. (2001) 27:171–81.
Montanari G. E. Post-sampling efficient prediction in large scale surveys. Int. Statist. Rev. (1987) 55:191–202.
Patterson H. Sampling on successive occasions with partial replacement of units. J. R. Statist. Soc. (1950) B 12:241–55.
Pires A., Branco J. Partial influence functions. J. Mult. Anal. (2002) 83:451–68.[CrossRef]
Place D. Calcul de la précision des estimations longitudinales dans l'enquête emploi en continu. In: Méthodes de Sondage: Applications aux Enquêtes Longitudinales, à la Santé, aux Enquêtes Electorales et aux Enquêtes dans les Pays en Développement—Guibert P., Haziza D., Ruiz-Gazen A., Tillé Y., eds. (2008) Paris: Dunod. 53–57.
Rao J. N. K., Wu C. F. J., Yue K. Some recent works on resampling methods for complex surveys. Survey Methodol. (1992) 18:209–17.
Reid N. Influence functions for censored data. Ann. Statist. (1981) 9:78–92.[CrossRef]
Särndal C. E., Swenson B., Wretman J. H. Model Assisted Survey Sampling (1992) New York: Springer.
Singh A. C., Kennedy B., Wu S. Regression composite estimation for the Canadian labour force survey with a rotating panel design. Survey Methodol. (2001) 27:33–44.
Tam S. M. On covariances from overlapping samples. Am. Statistician (1984) 38:288–9.[CrossRef]
von Mises R. On the asymptotic distribution of differentiable statistical functions. Ann. Math. Statist. (1947) 18:309–48.[CrossRef]
Wolter K. Introduction to Variance Estimation (2007) New York: Springer.
Wu C. Combining information from multiple surveys through the empirical likelihood method. Can. J. Statist. (2003) 32:15–26.[CrossRef]
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||