Dimension estimation in sufficient dimension reduction: A unifying approach

Abstract

a b s t r a c tSufficient Dimension Reduction (SDR) in regression comprises the estimation of the dimension of the smallest (central) dimension reduction subspace and its basis elements. For SDR methods based on a kernel matrix, such as SIR and SAVE, the dimension estimation is equivalent to the estimation of the rank of a random matrix which is the sample based estimate of the kernel. A test for the rank of a random matrix amounts to testing how many of its eigen or singular values are equal to zero. We propose two tests based on the smallest eigen or singular values of the estimated matrix: an asymptotic weighted chi-square test and a Wald-type asymptotic chi-square test. We also provide an asymptotic chi-square test for assessing whether elements of the left singular vectors of the random matrix are zero. These methods together constitute a unified approach for all SDR methods based on a kernel matrix that covers estimation of the central subspace and its dimension, as well as assessment of variable contribution to the lower-dimensional predictor projections with variable selection, a special case. A small power simulation study shows that the proposed and existing tests, specific to each SDR method, perform similarly with respect to power and achievement of the nominal level. Also, the importance of the choice of the number of slices as a tuning parameter is further exhibited.

A Unified Framework for Dimension Reduction in Forecasting

“…., it can also be shown that their sample means are approximately normally distributed for large T (see Appendix B). Under the same assumptions we can then obtain that M h is asymptotically normal following similar arguments as Bura and Yang (2011) [17] who derived the asymptotic distribution of M when the data are i.i.d. draws from the joint distribution of (y, x).…”

“…It is based on the results of Section 3.1 and uses a sample counterpart to Σ −1 Var (E (x|y)). 17 The name derives from using the inverse regression of x on the sliced response y to estimate the reduction. For a univariate y, the method is particularly easy to implement, SIR's step functions being a simple nonparametric approximation to E(x|y).…”

Inverse Regression for the Wiener Class of Systems

“…The first two methods are based on the results in Theorem 1 while the third method relates to Theorem 2. In the Step 3 of Algorithm 1, one needs to determine an appropriate value for the dimension d. This can be done in several ways, either by the use of some test statistic [Bura and Yang, 2011] or via a straightforward inspection of a plot of the eigenvalues.…”

Sliced Inverse Regression for the Identification of Dynamical Systems

“…If the nonlinearity in (1) is even about the origin, then the conditional expectation (4) will be zero and no information regarding the projection will be present in the data. To remedy this limitation, several methods involving second order moments have been proposed in the literature, see, for example, Cook and Weisberg [1991], Li and Wang [2007], Bura and Yang [2011]. The evaluation of some of the above methods on systems from the Wiener class can be found in Lyzell and Enqvist [2011].…”

“…In the one-dimensional case, it is well known that if ϕ(t) is elliptically distributed, the linear least-squares estimate of B is consistent [Bussgang, 1952]. In this paper, we are going to consider a related method called sliced inverse regression (SIR) [Li, 1991], which has had a considerable influence in the field of dimension reduction in the statistical community [see, for example, Bura and Cook, 2001, Li and Wang, 2007, Bura and Yang, 2011.…”