We consider the following calibration problem given n pairs (x, Y) from a linear regression with normal residuals, estimate x for a given Y. The mean of the ‘naive’ estimate does not exist. Suitably modified it has bias ~ n −1. With one correction term the bias is reduced to an almost exponentially small amount. The estimates require knowing a lower bound for the absolute value of the slope.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Azzalini A (1985) A class of distributions include the normal ones. Scand J Stat 12: 171–178
Brown PJ (1994) Measurement, regression and calibration. Oxford University Press, Oxford
Carroll RJ, Ruppert D, Stefanski LA (1995) Measurement error in nonlinear models. Chapman and Hall, London
Carroll RJ, Ruppert D, Stefanski LA, Crainiceanu CM (2006) Measurement error in nonlinear models: a modern perspective. 2nd edn. Chapman and Hall/CRC, Boca Raton
Chen K, Fan JQ, Jin ZZ (2008) Design-adaptive minimax local linear regression for longitudinal/clustered data. Stat Sin 18: 515–534
Chen Q, Giles DE (2011) Finite-sample properties of the maximum likelihood estimator for the binary logit model with random covariates. Stat Pap. doi:10.1007/s00362-010-0348-z
Cheng MY, Peng L, Wu JS (2007) Reducing variance in univariate smoothing. Ann Stat 35: 522–542
Choi E, Hall P (1998) On bias reduction in local linear smoothing. Biometrika 85: 333–345
He H, Huang LS (2009) Double-smoothing for bias reduction in local linear regression. J Stat Plan Inference 139: 1056–1072
Osborne C (1991) Statistical calibration: a review. Int Stat Rev 59: 309–336
Patriota AG, Lemonte AJ, Bolfarine H (2011) Improved maximum likelihood estimators in a heteroskedastic errors-in-variables model. Stat Pap 52: 455–467
Tanizaki H, Hamori S, Matsubayashi Y (2006) On least-squares bias in the AR(p) models: bias correction using the bootstrap methods. Stat Pap 47: 109–124
Withers CS (1987) Bias reduction by taylor series. Commun Stat Theory Methods 16: 2369–2384
Withers CS, Nadarajah S (2010) Estimates for inverse powers of a normal mean. Electron J Stat (in press)
Yanagihara H (2006) Corrected version of AIC for selecting multivariate normal linear regression models in a general nonnormal case. J Multivar Anal 97: 1070–1089
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Withers, C.S., Nadarajah, S. Calibration with low bias. Stat Papers 54, 371–379 (2013). https://doi.org/10.1007/s00362-012-0433-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-012-0433-6