Optimal transformation: A new approach for covering the central subspace

Abstract

voir prépublication : hal-00598422International audienceThis paper studies a general family of methods for sufficient dimension reduction (SDR) called the test function (TF), based on the introduction of a nonlinear transformation of the response. By considering order 1 and 2 conditional moments of the predictors given the response, we distinguish two classes of methods. The optimal members of each class are calculated with respect to the asymptotic mean squared error between the central subspace (CS) and its estimate. Moreover the theoretical background of TF is developed under weaker conditions than the existing methods. Accordingly, simulations confirm that the resulting methods are highly accurate

An Empirical Process View of Inverse Regression

“…Then, conditionally on the sample, and ; or (ii) and .Remark (on the coverage property) A comparison between the spaces generated by CUME and SIR is relevant to highlight the differences between continuous and discrete methods. The space which is estimated by SIR is Under the conditions of theorem 3 in Portier & Delyon (), for H sufficiently large, . It follows that as H increases, SIR eventually estimates the whole subspace.…”

“…Throughout the paper, we will assume that the central subspace generated by β 0 is unique. This assumption holds whenever X has a density (Portier & Delyon, , theorem 1). To improve clarity in the remainder of the paper, we introduce the standardized predictor Z =Σ −1/2 ( X − E X ), where Σ=Var( X ).…”

“…Although the conditional quantities E ( Z ∣ Y ) and Var( Z ∣ Y ) are not estimated at a parametric rate , that is, at rate ‘root n ’, it is possible to estimate Var( E ( Z ∣ Y )) or other moments of these quantities at a parametric rate. On the other hand, one might consider a constant number of slices, that is, the length of the slices does not converge to zero (Cook & Weisberg, ; Cook & Ni, ; Li & Wang, ; Portier & Delyon, ). For the first moments methods, one might argue in favour of the latter approach, because for any bounded measurable function ψ .…”

“…For the first moments methods, one might argue in favour of the latter approach, because for any bounded measurable function ψ . Here, the subspace E c will eventually be recovered if the number of functions ψ is large (Portier & Delyon, , theorem 3). Going further, a natural idea is to consider estimation of E c when ψ is not necessarily a slicing but another class of functions.…”

“…Going further, a natural idea is to consider estimation of E c when ψ is not necessarily a slicing but another class of functions. Examples can be found in Yin & Cook (), where the use of polynomial functions ψ is discussed, and in Portier & Delyon (), where the optimal choice of ψ within a Hilbert space is considered (see also Bernard‐Michel et al () for the use of basis functions). In Zhu et al (), a non‐countable class of functions is considered: the indicators of the sets { Y ≤ t } for .…”