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Biometrika 1999 86(4):831-842; doi:10.1093/biomet/86.4.831
© 1999 by Biometrika Trust
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On extended partially linear single-index models

Y XiaA1, H TongA2 and WK LiA3

Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong PRC A1 E-mail: ycxia@hkusua.hku.hk A2 htong@hku.hk A3 hrntlwk@hku.hk

Aiming to explore the relation between the response y and the stochastic explanatory vector variable X beyond the linear approximation, we consider the single-index model, which is a well-known approach in multidimensional cases. Specifically, we extend the partially linear single-index model to take the form y=ßT0X + {phi}({theta}T0X) + {epsilon}, where {epsilon} is a random variable with E{epsilon}=0 and var({epsilon})={sigma}2, unknown, ß0 and {theta}0 are unknown parametric vectors and {phi}(.) is an unknown real function. The model is also applicable to nonlinear time series analysis. In this paper, we propose a procedure to estimate the model and prove some related asymptotic results. Both simulated and real data are used to illustrate the results.

Key Words: alpha-mixing; kernel smoothing; nonlinear time series; partially linear model; single-index model


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