The cross-section distribution of U.S. import prices exhibits some of the fat-tailed characteristics that are well documented for the cross-section distribution of U.S. consumer prices. This suggests that limited-influence estimators of core import price inflation might outperform headline or traditional measures of core import price inflation. We examine whether limited influence estimators of core import price inflation help forecast overall import price inflation. They do not. However, limited influence estimators of core import price inflation do seem to have some predictive power for headline consumer price inflation in the medium term.