# zbMATH — the first resource for mathematics

On extended partially linear single-index models. (English) Zbl 0942.62109
Summary: 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 $y=\beta^T_0 X+\varphi (\theta^T_0 X)+ \varepsilon,$ where $$\varepsilon$$ is a random variable with $$E\varepsilon=0$$ and $$\text{var} (\varepsilon)= \sigma^2$$, unknown, $$\beta_0$$ and $$\theta_0$$ are parametric vectors and $$\varphi(\cdot)$$ is an unknown real function. The model is also applicable to nonlinear time series analysis. 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.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 65C60 Computational problems in statistics (MSC2010)
Full Text: