Statistical estimation in partially linear single-index models with error-prone linear covariates

Abstract

This article considers a class of partially linear single-index models when some linear covariates are not observed, but their ancillary variables are available. This model can avoid the ‘curse of dimensionality’ in multivariate nonparametric regressions, and it contains many existing statistical models such as the partially linear model (Engle, R.F., Granger, W. J., Rice, J., and Weiss, A. (1986), ‘Semiparametric Estimates of the Relation Between Weather and Electricity Sales’, Journal of The American Statistical Association, 80, 310–319), the single-index model (Härdle, W., Hall, P., and Ichimura, H. (1993), ‘Optimal Smoothing in Single-Index Models’, The Annals of Statistics, 21, 157–178), the partially linear errors-in-variables model (Liang, H., Härdle, W., and Carroll, R.J. (1999), ‘Estimation in a Semi-parametric Partially Linear Errors-in-Variables Model’, The Annals of Statistics, 27, 1519–1535), the partially linear single-index measurement error model (Liang, H., and Wang, N. (2005), ‘Partially Linear Single-Index Measurement Error Models’, Statistica Sinica, 15, 99–116), and so on as special examples. In this article, an estimation procedure for the unknowns of the proposed models is proposed, and asymptotic properties of the corresponding estimators are derived. Finite sample performance of the proposed methodology is assessed by Monte Carlo simulation studies. A real example is also given to illustrate the proposed procedures.