ABSTRACT
In the development of Chapters 2 through 6, the Sufficient Dimension Reduction problem is posed as reducing the dimension of x in the conditional distribution of Y|X. However, in many applications, such as nonparametric regression and single index models (Ichimura, 1993), our primary interest of estimation is the conditional mean E(Y|X) rather than entire conditional distribution of F Y | X $ F _{\scriptscriptstyle Y|X} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/inline-math8_1.tif"/> . In this case, it is beneficial to focus on the conditional mean when conducting dimension reduction, which can result in further reduction of dimension. In fact, some earlier Sufficient Dimension Reduction methods, such as Ordinary Least Squares and Principal Hessian Directions (Li and Duan, 1989; Li, 1992) targeted the conditional mean, even though this was not realized or formulated explicitly. In this chapter we lay out the theoretical framework and some techniques for estimating Sufficient Dimension Reduction for conditional mean.
