Regression methods for high dimensional multicollinear data

To compare their performance on high dimensional data, several regression methods are applied to data sets in which the number of exploratory variables greatly exceeds the sample sizes. The methods are stepwise regression, principal components regression, two forms of latent root regression, partial least squares, and a new method developed here. The data are four sample sets for which near infrared reflectance spectra have been determined and the regression methods use the spectra to estimate the concentration of various chemical constituents, the latter having been determined by standard chemical analysis. Thirty-two regression equations are estimated using each method and their performances are evaluated using validation data sets. Although it is the most widely used, stepwise regression was decidedly poorer than the other methods considered. Differences between the latter were small with partial least squares performing slightly better than other methods under all criteria examined, albeit not by a statistically significant amount.     

Bounds and approximations for pairwise comparisons of independent multinormal means

Properties and relationships of some commonly used probability bounds, along with other recently developed bounds and approximations, are evaluated for their performance with pairwise comparisons. The comparisons are of independent sample means obtained from normal random variables with a common variance. Computational methods are presented and numerical results are used to further evaluate the performance of the bounds.