A Semiparametric Approach to Dimension Reduction

Ma, Yanyuan; Zhu, Liping

doi:10.1080/01621459.2011.646925

Cited by 190 publications

(204 citation statements)

References 35 publications

(49 reference statements)

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“…For example, see Cook & Weisberg (save, 1991), Li (phd, 1992), Yin & Cook (Cov k , 2002), Xia et al (mave, 2002), the seminal papers of Ma & Zhu (2012, 2013a and for a detailed review see Cook (1998b) or Cook & Weisberg (1999). All of these methods consider only a univariate response and thus, dimension reduction is performed only on the predictor variables.…”

Section: Introductionmentioning

confidence: 97%

The Dual Central Subspaces in dimension reduction

Iaci

Yin

Zhu

2016

Journal of Multivariate Analysis

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Existing dimension reduction methods in multivariate analysis have focused on reducing sets of random vectors into equivalently sized dimensions, while methods in regression settings have focused mainly on decreasing the dimension of the predictor variables. However, for problems involving a multivariate response, reducing the dimension of the response vector is also desirable and important. In this paper, we develop a new concept, termed the Dual Central Subspaces (DCS), to produce a method for simultaneously reducing the dimensions of two sets of random vectors, irrespective of the labels predictor and response. Different from previous methods based on extensions of Canonical Correlation Analysis (CCA), the recovery of this subspace provides a new research direction for multivariate sufficient dimension reduction. A particular model-free approach is detailed theoretically and the performance investigated through simulation and a real data analysis.

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Section: Introductionmentioning

confidence: 97%

The Dual Central Subspaces in dimension reduction

Iaci

Yin

Zhu

2016

Journal of Multivariate Analysis

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“…For example, in the dimension reduction models, when covariates are random samples from an elliptical distribution family, great computational simplification occurs due to a robustness property (Ma & Zhu, 2012). However, transformation on the original covariates is almost always needed to achieve ellipticity.…”

Section: Semiparametric Models In Transformation Constructionmentioning

confidence: 99%

Discussion

2014

Int Statistical Rev

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I congratulate Chris Jones on his excellent work on summarizing various ways of generating univariate distribution families. Broadly speaking, families of univariate random variable distributions are often generated in two ways, through special selection mechanism as described in family 1 and through transformation as described in families 2, 3, and 4. Because of their close adaptation to the data generation procedure, I consider these modelling procedures to be natural and interpretable and so is the resulting distribution families. It is also quite amazing that these seemingly simple procedures can generate such rich families of parametric distributions. To some degree, parametric families of distributions form a rather mature field in statistics. One can envision new distribution construction only if very special application arises that calls for such needs. The same applies to parameter estimation and inference, in that a maximum likelihood estimation procedure usually yields the efficient estimator under suitable regularity conditions and its large sample properties are generally well understood. Because of these observations, in the following, I mainly concentrate the discussion on various extensions of the univariate parametric distribution families. Some of these extensions are well studied, whereas others are still waiting to be explored.

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“…On the other hand, as illustrated in some existing literature, the low structural dimension might be sufficient for many practical problems. For example, Cook (1998b) analyzed the Motor Octane data and selected the structural dimension as 1 by his proposed chi-square test; Xia et al (2002) chose the dimension as 2 for the Hitter's Salary data using the cross-validation; Ma and Zhu (2012) used the bootstrap procedure to determine the dimension as 1 for the Employee's Salary data from the fifth National Bank of Springfield. Zhu et al (2011) also pointed out that, by the purpose of dimension reduction, the structural dimension is generally assumed to be small and takes values 1, 2 or 3.…”

Section: Dimension-reduction-based Imputation For Sir (Dri-sir)17mentioning

confidence: 99%

Sufficient Dimension Reduction under Dimension-reduction-based Imputation with Predictors Missing at Random

Yang¹,

Wang²

2019

STAT SINICA

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Abstract:In some practical problems, a subset of predictors are frequently subject to missingness, especially when the dimension of the predictors is high. For this case, the standard sufficient dimension reduction (SDR) methods cannot be applied directly to avoid the curse of dimensionality. A dimension-reductionbased imputation method is developed in this article such that any of spectraldecomposition-based SDR methods for full data is applicable to the case of predictors missing at random. The sliced inverse regression (SIR) is used to illustrate this procedure. The proposed dimension-reduction-based imputation estimator of the candidate matrix for SIR, termed as DRI-SIR estimator, is asymptotically normal under some mild conditions and hence the resulting estimator of the central subspace is root n consistent. The finite sample performance of the proposed method is evaluated through comprehensive simulations and a real data set is analyzed for illustration.

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A Semiparametric Approach to Dimension Reduction

Cited by 190 publications

References 35 publications

The Dual Central Subspaces in dimension reduction

The Dual Central Subspaces in dimension reduction

Discussion

Sufficient Dimension Reduction under Dimension-reduction-based Imputation with Predictors Missing at Random

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