Envelopes for elliptical multivariate linear regression

Abstract

We incorporate the idea of reduced rank envelope [7] to elliptical multivariate linear regression to improve the efficiency of estimation. The reduced rank envelope model takes advantage of both reduced rank regression and envelope model, and is an efficient estimation technique in multivariate linear regression. However, it uses the normal log-likelihood as its objective function, and is most effective when the normality assumption holds. The proposed methodology considers elliptically contoured distributions and it incorporates this distribution structure into the modeling. Consequently, it is more flexible and its estimator outperforms the estimator derived for the normal case. When the specific distribution is unknown, we present an estimator that performs well as long as the elliptically contoured assumption holds.

Envelopes: A new chapter in partial least squares regression

“…32,33 Chemometricians may find results on tensor envelope partial least squares regression 34 to be of particular interest. Forzani and Su 35 recently extended the normal likelihood theory underlying envelopes to elliptically contoured distributions. Each of these and others demonstrate a potential for envelope methodology to achieve reduction in estimative and predictive variation beyond that attained by standard methods, sometimes by amounts equivalent to increasing the sample size many times over.…”

Review of Dimension Reduction Methods

“…There is also the partial least square (PLS) [331] [332], sufficient component analysis (SCA) [333], kernel dimension reduction (KDR) [334]. Other methods include, but are not limited to, the method proposed by [335] for exponential family predictors and the methods suggested by [336] with exponential family inverse predictors and the likelihood based dimension reduction method which was proposed by [337]. The limitation of most of the SDR techniques however, is that they require linearity condition which includes SIR and SAVE [338] or the constant variance condition [320] [321] or even both to hold for some techniques, which is practically difficult to verify.…”

A Review of Envelope Models

“…In such cases, it is better to develop an envelope estimator based on the specific parametric structure of the model, which is normally more efficient than the estimator produced from the general procedure. Besides the GLM example, other examples on using the specific parametric structure to develop an envelope estimator can be found in Cook et al (2015), Rekabdarkolaee et al (2019) and Forzani and Su (2019).…”

“…Cook and Zhang (2015) derived the simultaneous envelope model which performs dimension reduction on both and to achieve further efficiency gains than the response envelope model or predictor envelope model. Foranzi and Su (2019) applied the envelope model to elliptical multivariate linear regression to improve estimation efficiency gains. This model allows for heteroscedastic errors without requiring any groupings of the data.…”