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Envelope methods

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Abstract An envelope is a relatively new construct for decreasing estimative and predictive variation relative to standard methods in multivariate statistics, sometimes by amounts equivalent to increasing the sample size many times over. Essentially a form of targeted dimension reduction that is descendent from sufficient dimension reduction, an envelope inherits its underlying philosophy from Fisher's notion of sufficient statistics. The initial development of envelope methods took place largely in the context of the multivariate linear model, resulting in response envelopes for response reduction, predictor envelopes for predictor reduction, simultaneous envelopes for response and predictor reduction and partial envelopes for specialized considerations, each demonstrating a potential for substantial reduction in estimative variation. These advances demonstrated that there are close connections between envelopes and some standard multivariate methods like partial least squares regression and canonical correlations. Subsequently, envelope methodology has been adapted and extended to diverse areas, including envelopes for regressions with matrix and tensor‐valued responses, envelopes for spatial statistics, quantile envelopes for quantile regression, Bayesian response envelopes. Sparse versions of response and predictor envelopes have also been developed. More generally, there is also envelope methodology for reducing the variation in any asymptotically normal vector‐valued estimator. These advances have opened a new chapter in multivariate statistics, allowing variation that is material to the goals of an analysis to be separated effectively from immaterial variation that serves only to confound estimation and prediction. This article is categorized under: Statistical Models > Multivariate Models Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Dimension Reduction
Schematic illustration of response envelopes for the regression of a bivariate response Y = (Y1, Y2)T on a binary predictor. The distributions of Y ∣ X are represented by the gray and blue ellipses. The red solid and dashed lines represent the envelope and its orthogonal complement
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Statistical and Graphical Methods of Data Analysis > Multivariate Analysis
Statistical and Graphical Methods of Data Analysis > Dimensional Reduction
Statistical Models > Multivariate Models

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