On the two-phase framework for joint model and design-based inference
Susana Rubin-Bleuer and Ioana Schiopu Kratina
Source: Ann. Statist. Volume 33, Number 6 (2005), 2789-2810.
Abstract
We establish a mathematical framework that formally validates the two-phase “super-population viewpoint” proposed by Hartley and Sielken [Biometrics 31 (1975) 411–422] by defining a product probability space which includes both the design space and the model space. The methodology we develop combines finite population sampling theory and the classical theory of infinite population sampling to account for the underlying processes that produce the data under a unified approach. Our key results are the following: first, if the sample estimators converge in the design law and the model statistics converge in the model, then, under certain conditions, they are asymptotically independent, and they converge jointly in the product space; second, the sample estimating equation estimator is asymptotically normal around a super-population parameter.
Primary Subjects: 62F12
Secondary Subjects: 62D05
Keywords: Joint design and model-based inference
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1140191673
Digital Object Identifier: doi:10.1214/009053605000000651
Mathematical Reviews number (MathSciNet):
MR2253102
Zentralblatt MATH identifier:
1084.62020
References
Billingsley, P. (1979). Probability and Measure. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR0534323
Zentralblatt MATH:
0411.60001
Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR1324786
Zentralblatt MATH:
0822.60002
Binder, D. A. (1983). On the variances of asymptotically normal estimators from complex surveys. Internat. Statist. Rev. 51 279--292.
Mathematical Reviews (MathSciNet):
MR0731144
Binder, D. and Roberts, G. (2003). Design-based and model-based methods for estimating model parameters. In Analysis of Survey Data (R. L. Chambers and C. J. Skinner, eds.) 29--48. Wiley, Chichester.
Mathematical Reviews (MathSciNet):
MR1978842
Digital Object Identifier: doi:10.1002/0470867205.ch3
Chow, Y. S. and Teicher, H. (1997). Probability Theory. Independence, Interchangeability, Martingales, 3rd ed. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1476912
Zentralblatt MATH:
0891.60002
Fuller, W. A. (1975). Regression analysis for sample surveys. Sankhyā Ser. C 37 117--132.
Godambe, V. P. and Thompson, M. E. (1986). Parameters of super-population and survey population: Their relationships and estimation. Internat. Statist. Rev. 54 127--138.
Mathematical Reviews (MathSciNet):
MR0962931
Hájek, J. (1960). Limiting distributions in simple random sampling from a finite population. Publ. Math. Inst. Hungarian Acad. Sci. 5 361--374.
Mathematical Reviews (MathSciNet):
MR0125612
Hájek, J. (1964). Asymptotic theory of rejective sampling with varying probabilities from a finite population. Ann. Math. Statist. 35 1491--1523.
Mathematical Reviews (MathSciNet):
MR0178555
Hájek, J. (1981) (assembled after his death by Václav Dupac). Sampling from a Finite Population. Dekker, New York.
Mathematical Reviews (MathSciNet):
MR0627744
Hartley, H. O. and Sielken, R. L. (1975). A ``superpopulation viewpoint'' for finite population sampling. Biometrics 31 411--422.
Mathematical Reviews (MathSciNet):
MR0386084
Isaki, C. and Fuller, W. (1982). Survey design under the regression superpopulation model. J. Amer. Statist. Assoc. 77 89--96.
Mathematical Reviews (MathSciNet):
MR0648029
Kish, L. (1995). Survey Sampling. Wiley, New York.
Zentralblatt MATH:
0853.62013
Korn, E. L. and Graubard, B. I. (1998). Variance estimation for super-population parameters. Statist. Sinica 8 1131--1151.
Mathematical Reviews (MathSciNet):
MR1666241
Krewski, D. and Rao, J. N. K. (1981). Inference from stratified samples: Properties of linearization, jackknife and balanced repeated replication methods. Ann. Statist. 9 1010--1019.
Mathematical Reviews (MathSciNet):
MR0628756
Molina, E. A., Smith, T. M. F. and Sugden, R. A. (2001). Modelling overdispersion for complex survey data. Internat. Statist. Rev. 69 373--384.
Pfeffermann, D. and Sverchkov, M. (1999). Parametric and semiparametric estimation of regression models fitted to survey data. Sankhyā Ser. B 61 166--186.
Mathematical Reviews (MathSciNet):
MR1720710
Rodríguez, J. E. (2001). A probabilistic framework for inference in finite population sampling. Proc. Section on Survey Research Methods of the American Statistical Association, Aug. 5--9, 2001. Available at www.amstat.org/sections/srms/proceedings/y2001/proceed/00468.pdf.
Rosén, B. (1972). Asymptotic theory for successive sampling with varying probabilities without replacement. I, II. Ann. Math. Statist. 43 373--397, 748--776.
Mathematical Reviews (MathSciNet):
MR0321223
Rosén, B. (1997). On sampling with probability proportional to size. J. Statist. Plann. Inference 62 159--191.
Mathematical Reviews (MathSciNet):
MR1468159
Digital Object Identifier: doi:10.1016/S0378-3758(96)00186-3
Rubin-Bleuer, S. (1998). Inference for parameters of the super-population, Part I. Research sabbatical report, internal report, Statistics Canada.
Rubin-Bleuer, S. (2000). Some issues in the analysis of complex survey data. Statistics Canada Series, Methodology Branch, Business Survey Methods Division, BSMD-20-001E.
Rubin-Bleuer, S. (2001). A test for survival distributions using data from a complex sample. Proc. Survey Methods Section, SSC Annual Meeting, 2001 103--110.
Rubin-Bleuer, S. (2003). An approximation of the partial likelihood score in a joint design-model space. Proc. Survey Methods Section, SSC Annual Meeting, 2003 CDROM 37--46.
Rubin-Bleuer, S. and Schiopu Kratina, I. (2000). Some issues in the analysis of complex survey data. Proc. Section on Survey Research Methods of the American Statistical Association 734--739. Available at www.amstat.org/sections/srms/proceedings/papers/2000_124.pdf.
Rubin-Bleuer, S. and Schiopu Kratina, I. (2001). A probabilistic set-up for model and design-based inference. Proc. Section on Survey Research Methods of the American Statistical Association, Aug. 5--9, 2001. Available at www.amstat.org/sections/srms/proceedings/y2001/proceed/00469.pdf.
Rubin-Bleuer, S. and Schiopu Kratina, I. (2002). On the two-phase framework for joint model and design-based inference. Technical Report Series, Laboratory for Research in Probability and Statistics, No. 382, Carleton Univ.--Univ. Ottawa.
Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1140409
Scott, A. J. (1977). On the problem of randomization in survey sampling. Sankhyā Ser. C 39 1--9.
Mathematical Reviews (MathSciNet):
MR0456593
Digital Object Identifier: doi:10.1016/0022-2496(77)90039-6
Yuan, K.-H. and Jennrich, R. (1998). Asymptotics of estimating equations under natural conditions. J. Multivariate Anal. 65 245--260.
Mathematical Reviews (MathSciNet):
MR1625893
Digital Object Identifier: doi:10.1006/jmva.1997.1731
Yung, W. and Rao, J. N. K. (2000). Jackknife variance estimation under imputation for estimators using poststratification information. J. Amer. Statist. Assoc. 95 903--915.
Mathematical Reviews (MathSciNet):
MR1804447
The Annals of Statistics
