# Why not take the mode of a bootstrap distribution?

Say that I had a bootstrap distribution, can I return the mode as a point estimate? It's right-skewed, so the mean does not accurately summarize the distribution.

• What is right skewed? – Richard Hardy Jul 30 '16 at 16:41
• You take a sharp skewer and then pull the right side away from the center. – DWin Jul 30 '16 at 21:24

A pragmatic answer is that a meaningful mode is surprisingly hard to estimate in real data. Say we get all distinct point estimates in the bootstrap samples, then all estimated coefficients are the mode, which is probably not what we want. We could smooth the distribution, but then the number of modes is dependent on the smoothing parameter, etc. etc.

Others here can give you a technical answer as to exactly why we take the mean and not the mode. I expect it has to do with the fact that the mean is the expected value.

It's right-skewed, so the mean does not accurately summarize the distribution.

is not really correct, although I can certainly see where you could get that idea. The problem with the mean for skewed distributions is not that it is not an accurate summary but, rather, that for most purposes it is the wrong summary - that is, it is a good estimate of the wrong thing.

The classic example is income. In most countries, income is quite skewed so, if you want a one-number summary, the median corresponds better than the mean to what most people mean by "summary"; but for different purposes, the mean could be best, or the mode, or a trimmed mean, or even no measure at all.

• But why would you report the mean of the bootstrap samples? Should you not report the statistic applied on the original sample plus some measure of variability (e.g. the standard deviation) taken from the bootstrap samples? – Gabriel Nov 5 '19 at 13:58
• You could report the median. As to why you don't report the mode, see Gabriel's answer – Peter Flom Nov 5 '19 at 14:09
• But the median is a just another point estimator from the bootstrap samples. It was my understanding that you don't report the mean, median, or mode, you report the statistic obtained with the actual sample plus some measure of variability taken from the bootstrap sample (stats.stackexchange.com/questions/71357/…) Is this not true? (BTW: I think you meant Maarten's answer?) – Gabriel Nov 5 '19 at 15:21
• Yes, I meant @maarten's answer. But ... well, I guess then the answer is "don't look for modes via bootstrap". – Peter Flom Nov 6 '19 at 12:32