I've fit a regression model and diagnostic tests are showing some values have high Cook's D as well as high DFBETAS on my parameter of interest. The effect does not cross the traditional threshold for statistical significance but it is in the predicted direction. I could just remove these influential cases and see what happens (doing this pushes the further in the predicted direction), but I wonder if leaving them in and bootstrapping the parameter to dilute the influence of any one data point would be a more conservative, less potentially arbitrary approach. Indeed, bootstrapping the parameter with 5000 resamples and calculating bias corrected and accelerated 95% confidence intervals, the CI excludes zero.
I know there are discussions on whether to exclude outliers before bootstrapping (for example), but that's not my question here exactly. I'm wondering if this is a reasonable way to approach this issue and whether my interpretation of this pattern of results is safe. Can anyone point me to something, particularly a journal article, discussing using bootstrapping in this way?