Optimal non-diagonal-type estimators in linear regression under the prediction error sum of squares criterion

ABSTRACT

This article considers the notion of the non-diagonal-type estimator (NDTE) under the prediction error sum of squares (PRESS) criterion. First, the optimal NDTE in the PRESS sense is derived theoretically and applied to the cosmetics sales data. Second, we make a further study to extend the NDTE to the general case of the covariance matrix of the model and then give a Bayesian explanation for this extension. Third, two remarks concerned with some potential shortcomings of the NDTE are presented and an alternative solution is provided and illustrated by means of simulations.

KEYWORDS:

• Linear regression
• Modified non-diagonal-type estimator
• Non-diagonal-type estimator
• Prediction error sum of squares.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62J05

Acknowledgments

The authors are very grateful to the referees for valuable comments and constructive criticisms which result in the present version. Research supported in part by the National Natural Science Foundation of China (61374183, 10971097) and the Humanistic and Social Science Foundation of Ministry of Education of China (No. 12YJA630122).

Notes

1 Downloaded from http://gosset.wharton.upenn.edu/teaching/541/myers/PROB3.11.

2 Specifically, we first center and scale the data and then use the similar model (without the intercept term), because the three predictor variables have no the same unit. This is distinguished from the original model which uses a log-linear model.