Using the Bootstrap in Generalized Regression Estimation
James G. Booth1 and Alan H Welsh2
1. Department of Statistical Science, Cornell University, city/state?
2. Mathematical Sciences Institute, The Australian National University, Canberra
Regression adjustments are widely used for adjusting the sample mean of a variable of interest to account for deviations of the means of related auxiliary variables from their known population values. The adjustments produce estimators with variances smaller than that of the original sample mean. The method has a long history in the survey literature, and is closely related to covariance analysis in designed experiments and the control variates method used for variance reduction in Monte Carlo studies.
We discuss a generalized regression estimation procedure that can lead to much improved estimators of general population characteristics, such as quantiles, variances, and coefficients of variation. The key idea behind the approach is that the regression of an estimator of a target population parameter on the estimator of a (possibly vector-valued) auxiliary parameter is computed directly using the bootstrap. The method is quite general and requires minimal assumptions, the main ones being that the asymptotic joint distribution of the target and auxiliary parameter estimators is multivariate normal, and that the population values of the auxiliary parameters are known. The assumption on the asymptotic joint distribution implies that the relationship between the estimated target and the estimated auxiliary parameters is approximately linear with coefficients determined by their asymptotic covariance matrix. Use of the bootstrap to estimate these coefficients avoids the need for parametric distributional assumptions. First-order correct conditional confidence intervals based on asymptotic normality can be improved upon using quantiles of a conditional double bootstrap approximation to the distribution of the studentized target parameter estimate.
Alan Welsh holds the E.J. Hannan Professorship in Statistics at the Australian national University. He has contributed to statistical methodology and theory, including robustness and model selection in linear models, robustness and bootstrapping in linear mixed models, semiparametric estimation, modelling zero-inflated data, modelling compositional data and the analysis of data from sample surveys. He has written a major monograph on Aspects of Statistical Inference (Welsh 1996) and jointly authored another on Maximum Likelihood Estimation for Sample Survey Data (Chambers et al 2012). He was the Editor-in-Chief of the Australian and New Zealand Journal of Statistics from 2012-2015. Alan was elected a Fellow of the Australian Academy of Science in 2007, Fellow of the Institute of Mathematical Statistics in 1990, and Fellow of the American Statistical Association in 2002. He was awarded the Moran Medal of the Academy of Science for outstanding research in statistics by a person under the age of 40 in 1990, the Pitman Medal of the Statistical Society of Australia in 2012 and the E.A. (Alf) Cornish Award of the Australasian Region of the International Biometric Society in 2017.
This plenary address will be delivered in AH1 on Thursday 29 November at 2:00.