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June, 1991 Large Sample Theory of Estimation in Biased Sampling Regression Models. I
Peter J. Bickel, J. Ritov
Ann. Statist. 19(2): 797-816 (June, 1991). DOI: 10.1214/aos/1176348121


Biased sampling regression models were introduced by Jewell, generalizing the truncated regression model studied by Bhattacharya, Chernoff and Yang. If the independent variable takes on only a finite number of values (as does the stratum variable), we show: 1. That if the slope of the underlying regression model is assumed known, then the nonparametric maximum likelihood estimates of the distribution of the independent and dependent variables (a) can be calculated from ordinary $M$ estimates; (b) are asymptotically efficient. 2. How to construct $M$ estimates of the slope which are always $\sqrt n$ consistent, asymptotically Gaussian and are efficient locally, for example, if the error distribution is Gaussian. We support our asymptotics with a small simulation.


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Peter J. Bickel. J. Ritov. "Large Sample Theory of Estimation in Biased Sampling Regression Models. I." Ann. Statist. 19 (2) 797 - 816, June, 1991.


Published: June, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0742.62036
MathSciNet: MR1105845
Digital Object Identifier: 10.1214/aos/1176348121

Primary: 62G05
Secondary: 62G20

Keywords: $M$ estimates , Biased sampling , Nonparametric maximum likelihood , regression

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 2 • June, 1991
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