Open Access
May 2018 Asymptotics for the maximum sample likelihood estimator under informative selection from a finite population
Daniel Bonnéry, F. Jay Breidt, François Coquet
Bernoulli 24(2): 929-955 (May 2018). DOI: 10.3150/16-BEJ809


Inference for the parametric distribution of a response given covariates is considered under informative selection of a sample from a finite population. Under this selection, the conditional distribution of a response in the sample, given the covariates and given that it was selected for observation, is not the same as the conditional distribution of the response in the finite population, given only the covariates. It is instead a weighted version of the conditional distribution of interest. Inference must be modified to account for this informative selection. An established approach in this context is maximum “sample likelihood”, developing a weight function that reflects the informative sampling design, then treating the observations as if they were independently distributed according to the weighted distribution. While the sample likelihood methodology has been widely applied, its theoretical foundation has been less developed. A precise asymptotic description of a wide range of informative selection mechanisms is proposed. Under this framework, consistency and asymptotic normality of the maximum sample likelihood estimators are established. The theory allows for the possibility of nuisance parameters that describe the selection mechanism. The proposed regularity conditions are verifiable for various sample schemes, motivated by real problems in surveys. Simulation results for these examples illustrate the quality of the asymptotic approximations, and demonstrate a practical approach to variance estimation that combines aspects of model-based information theory and design-based variance estimation.


Download Citation

Daniel Bonnéry. F. Jay Breidt. François Coquet. "Asymptotics for the maximum sample likelihood estimator under informative selection from a finite population." Bernoulli 24 (2) 929 - 955, May 2018.


Received: 1 August 2015; Revised: 1 November 2015; Published: May 2018
First available in Project Euclid: 21 September 2017

zbMATH: 06778352
MathSciNet: MR3706781
Digital Object Identifier: 10.3150/16-BEJ809

Keywords: complex survey , pseudo-likelihood , stratified sampling , weighted distribution

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 2 • May 2018
Back to Top