Minimax Aspects of Bounded-Influence Regression

Abstract

Bounded-influence regression (in the sense of Hampel-Mallows-Krasker-Welsch) is examined critically from the point of view of finite-sample minimax robust estimation theory. The main conclusion is that the influence of position of very high leverage points—that is, of points where the diagonal element of the hat matrix H = X(X^TX)⁻¹ X^T exceeds a certain bound—should be cut down selectively. This bound should be chosen roughly between .2 and .5; for the upper of these values, the resulting estimate can be approximated very simply by scaling the residuals by their own estimated standard deviation. A conflict between decision theoretic (estimation) and data analytic (diagnostic) viewpoints is pointed out and briefly discussed.

Key Words:

• Robust regression
• Leverage points
• Hat matrix
• Minimax estimation
• Influence function
• Data analysis
• Diagnostics