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.
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Abstract |
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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