3.8 Conclusions and Further Discussion

In this chapter, we first extend the ADE method of Härdle and Stoker (1989) to the case of more than one e.d.r direction, i.e. the OPG method. This method has wide applicability in respect of designs for $ X$ and regression functions. To improve the accuracy of the estimation, we then propose our basic method, i.e. the MAVE method. Both theoretical analysis and simulations show that the MAVE method has many attractive properties. Different from all existing methods for the estimation of the directions, the MAVE estimators of the directions have a faster convergence rate than the corresponding estimators of the regression function. Based on the faster convergence rate, a method for the determination of the number of e.d.r directions is proposed. The MAVE method can also be extended easily to more complicated models. It does not require strong assumptions on the design of $ X$ and the regression functions.

Following the basic idea, we proposed the iMAVE method, which is closely related to the SIR method. In our simulations, the iMAVE method has better performance than the SIR method. The refined kernel based on the determination of the number of the directions can further improve the estimation accuracy of the directions. Our simulations show that substantial improvements can be achieved.

Unlike the SIR method, the MAVE method is well adapted to time series. Furthermore, all of our simulations show that the MAVE method has much better performance than the SIR method (even with $ r = 1 $ in the MAVE). This is rather intriguing because the SIR uses the one-dimensional kernel (for the kernel version) while the MAVE method uses a multi-dimensional kernel. However, because the SIR method uses $ y$ to produce the kernel weight, its efficiency will suffer from fluctuations in the regression function. The gain by using the $ y$-based one-dimensional kernel does not seem to be sufficient to compensate for the loss in efficiency caused by the fluctuation in the regression function. Further research is needed here.