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Recent results make the multivariate linear regression model much easier to use. This model has m ≥ 2 response variables. Results by Kakizawa (2009) and Su and Cook (2012) can be used to explain the large sample theory of the least squares estimator and of the widely used Wilks' Λ, Pillai's trace, and Hotelling Lawley trace test statistics. Kakizawa (2009) shows that these statistics have the same limiting distribution. This paper reviews these results and gives two theorems to show that the Hotelling Lawley test generalizes the usual partial F test for m = 1 response variable to m ≥ 1 response variables. Plots for visualizing the model are also given, and can be used to check goodness and lack of fit, to check for outliers and influential cases, and to check whether the error distribution is multivariate normal or from some other elliptically contoured distribution.

Selecting the most important predictor variables and achieving high prediction accuracy are the two main goals in Statistical Learning. In this work, we propose a new regularization method, HRLR (a Hybrid of Relaxed Lasso and Ridge Regression), where we use properties of both relaxed lasso and ridge regression. We also demonstrate the effectiveness of our method and the accuracy of the algorithm on simulated as well as real-world data. Simulation results suggest that the HRLR regression outperforms the existing well-known methods including lasso and Elastic net in many scenarios described, especially when the error distribution is non-normal. The proposed algorithm is implemented in R and links are provided in this paper along with the R functions used for the simulation.

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