A unified principled framework for resampling based on pseudo-populations: Asymptotic theory

In this paper, a class of resampling techniques for finite populations under complex sampling design is introduced. The basic idea on which it rests is a two-step procedure consisting in : (i) constructing a pseudo-population on the basis of sample data; (ii) drawing a sample from the predicted population according to an appropriate resampling design. From a logical point of view, this approach is essentially based on the plug-in principle by Efron, at the "sampling design level". Theoretical justifications based on large sample theory are provided. New approaches to construct pseudo-populations based on various forms of calibrations are proposed. Finally, a simulation study is performed.

[1]  Partha Lahiri,et al.  On the Impact of Bootstrap in Survey Sampling and Small-Area Estimation , 2003 .

[2]  A. Holmberg.  1998: A BOOTSTRAP APPROACH TO PROBABILITY PROPORTIONAL-TO-SIZE SAMPLING , 2002 .

[3]  A. V. D. Vaart.  Asymptotic Statistics: Delta Method , 1998 .

[4]  Yves Tillé,et al.  A Direct Bootstrap Method for Complex Sampling Designs From a Finite Population , 2011 .

[5]  D. Freedman,et al.  Some Asymptotic Theory for the Bootstrap , 1981 .

[6]  Mathématiques Appliquées.  MÉTHODES DE BOOTSTRAP EN POPULATION FINIE , 2008 .

[7]  P J McCarthy,et al.  The bootstrap and finite population sampling. , 1985, Vital and health statistics. Series 2, Data evaluation and methods research.

[8]  Guillaume Osier,et al.  Variance estimation for complex indicators of poverty and inequality using linearization techniques , 2009 .

[9]  A. Rosalsky.  A survey of limit laws for bootstrapped sums , 2003 .

[10]  Yves Tillé,et al.  Sampling Algorithms , 2011, International Encyclopedia of Statistical Science.

[11]  Y. Berger.  Asymptotic consistency under large entropy sampling designs with unequal probabilities , 2011 .

[12]  P. Sen,et al.  Large sample methods in statistics , 1993 .

[13]  R. Sundaram.  A First Course in Optimization Theory: Optimization in ℝ n , 1996 .

[14]  D. Pfeffermann.  The Role of Sampling Weights when Modeling Survey Data , 1993 .

[15]  B. Efron,et al.  Second thoughts on the bootstrap , 2003 .

[16]  Y. Berger.  Variance estimation with Chao's sampling scheme , 2005 .

[17]  P. Massart.  The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality , 1990 .

[18]  L. Bondesson,et al.  Pareto Sampling versus Sampford and Conditional Poisson Sampling , 2006 .

[19]  Yves G. Berger,et al.  Rate of convergence to normal distribution for the Horvitz-Thompson estimator , 1998 .

[20]  C. Léger,et al.  A survey of bootstrap methods in finite population sampling , 2016 .

[21]  J. A. Vísek.  Asymptotic Distribution of Simple Estimate for Rejective, Sampford and Successive Sampling , 1979 .

[22]  Anders Lundqvist.  On the Distance Between Some πps Sampling Designs , 2007 .

[23]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[24]  F Mecatti,et al.  Comparing Recent Approaches for Bootstrapping Sample Survey Data: A First Step Toward a Unified Approach , 2013 .

[25]  Danny Pfeffermann,et al.  Prediction of finite population totals based on the sample distribution , 2004 .

[26]  Jun S. Liu,et al.  Weighted finite population sampling to maximize entropy , 1994 .

[27]  T. Postelnicu,et al.  Foundations of inference in survey sampling , 1977 .

[28]  P. Conti,et al.  Inference for Quantiles of a Finite Population: Asymptotic versus Resampling Results , 2015 .

[29]  A. Chatterjee.  Asymptotic properties of sample quantiles from a finite population , 2011 .

[30]  Pier Luigi Conti,et al.  On the Estimation of the Distribution Function of a Finite Population Under High Entropy Sampling Designs, with Applications , 2014, Sankhya B.

[31]  J. Hájek.  Asymptotic Theory of Rejective Sampling with Varying Probabilities from a Finite Population , 1964 .

[32]  J. Beaumont,et al.  On the Generalized Bootstrap for Sample Surveys with Special Attention to Poisson Sampling , 2012 .

[33]  C. Kraft.  Some conditions for consistency and uniform consistency of statistical procedures , 1955 .

[34]  S. Lahiri.  Resampling Methods for Dependent Data , 2003 .

[35]  Anton Grafström,et al.  Entropy of unequal probability sampling designs , 2010 .

[36]  Stéphan Clémençon,et al.  Empirical Processes in Survey Sampling with (Conditional) Poisson Designs , 2017 .

[37]  David Hinkley,et al.  Bootstrap Methods: Another Look at the Jackknife , 2008 .

[38]  P. Hall,et al.  Bootstrap Methods for Finite Populations , 1994 .

[39]  C. T. Isaki,et al.  Survey Design under the Regression Superpopulation Model , 1982 .

[40]  Stanislav Kolenikov,et al.  Resampling Inference with Complex Survey Data , 1996 .

[41]  Pier Luigi Conti,et al.  Statistical Matching Analysis for Complex Survey Data With Applications , 2016 .

[42]  S. Gross.  MEDIAN ESTIMATION IN SAMPLE SURVEYS , 2002 .

[43]  R. Sitter.  A resampling procedure for complex survey data , 1992 .

[44]  Daniela Marella,et al.  STRUCTURAL LEARNING FOR COMPLEX SURVEY DATA , 2017 .

[45]  Pier Luigi Conti,et al.  On the matching noise of some nonparametric imputation procedures , 2008 .

[46]  J. Hájek,et al.  Sampling from a finite population , 1982 .