Regularized multivariate analysis

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The theory of statistics has so far largely been dominated by asymptotic theories. However, we often have to deal with small sample sized data. In such situations, the maximum likelihood estimation (MLE), proven optimal for large samples, may not be optimal. I propose to use regularized estimation in such situations. The regularization means any processes that utilize prior knowledge in data analysis, and includes all such processes that are veriously called penalizing, smoothing, shrinking, soft constraining, etc. More specifically, I propose to incorporate a ridge type as well as other types of shrinkage estimation in redundancy analysis (reduced rank regression), multiple-set canonical correlation analysis as well as other techniques of multivariate analysis (MVA). This type of estimation is known to provide parameter estimates associated with smaller mean square errors (average discrepancies between true parameter values and the estimates) particularly in small samples. We demonstrate the effectiveness of regularization (shrinkage estimation) in these techniques, using both real data and simulation methods. We plan to write a series of user friendly computer programs and make them publicly available to make regularized MVA a routine part of multivariate data analysis.