Theoretical and empirical properties of model assisted decision-based regression estimators

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Jun Shao, Eric Slud, Yang Cheng, Sheng Wang, and Carma Hogue^{Note 1}

Abstract

In 2009, two major surveys in the Governments Division of the U.S. Census Bureau were redesigned to reduce sample size, save resources, and improve the precision of the estimates (Cheng, Corcoran, Barth and Hogue 2009). The new design divides each of the traditional state by government-type strata with sufficiently many units into two sub-strata according to each governmental unit’s total payroll, in order to sample less from the sub-stratum with small size units. The model-assisted approach is adopted in estimating population totals. Regression estimators using auxiliary variables are obtained either within each created sub-stratum or within the original stratum by collapsing two sub-strata. A decision-based method was proposed in Cheng, Slud and Hogue (2010), applying a hypothesis test to decide which regression estimator is used within each original stratum. Consistency and asymptotic normality of these model-assisted estimators are established here, under a design-based or model-assisted asymptotic framework. Our asymptotic results also suggest two types of consistent variance estimators, one obtained by substituting unknown quantities in the asymptotic variances and the other by applying the bootstrap. The performance of all the estimators of totals and of their variance estimators are examined in some empirical studies. The U.S. Annual Survey of Public Employment and Payroll (ASPEP) is used to motivate and illustrate our study.

Key Words:   Asymptotic normality; Bootstrap; Decision-based estimator; Probability proportional to size; Stratification; Variance estimation

Table of content

• 1 Introduction
• 2 Consistency and asymptotic normality
• 3 Variance estimation
• 4 Simulation results for results for H = 1
• Appendix
• Acknowledgements
• References

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Note

1. Jun Shao, Statistics Department University of Wisconsin, Madison WI , E-mail: shao@stat.wisc.edu; Eric Slud, Center for Statistical Research and Methodology, US Census Bureau, Washington DC and Mathematics Department, University of Maryland, College Park, MD, E-mail: eric.v.slud@census.gov; Yang Cheng, Demographic Statistical Methods Division, US Census Bureau, Washington DC, E-mail: yang.cheng@census.gov; Sheng Wang, Mathematica Policy Research, Princeton NJ, E-mail: swang@mathematica-mpr.com; and Carma Hogue, Governments Division, US Census Bureau, Washington DC, E-mail: carma.ray.hogue@census.gov.