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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.

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