How to get estimates for a population parameter when resampling distribution is skewed?

#1
Let's say I want to estimate some parameter p of a population. I have drawn a random sample S from it. One way of doing so is to resample N samples S_1, S_2, ..., S_N using, say, bootstrap or jackknife. I can then calculate that parameter's value for each S_i: call it p_i.

Then I look at the distribution of p_i's and find that it's skewed. In this scenario,
1. Is the resampling distribution of the parameter supposed to be symmetric, and therefore I should suspect something has gone wrong? In this case how do I get the point and interval estimates for p? My current thought is that it may be possible for even the resampling distribution to get skewed if the population distribution is heavily skewed to begin with
2. If it's acceptable for the resampling distribution to be skewed, should I use the mode instead of the mean or median as the point estimate? Usually we just say, take the mean of the resampling distribution as your point estimate. But in the presence of outliers, common wisdom says take the medium. But also for probability distributions, it's usually advised to take the mode as a point estimate. So which is it?
 

hlsmith

Less is more. Stay pure. Stay poor.
#2
What are you trying to do, get a measure of central tendency from a sample?

The bootstrap distribution of whatever estimand will be symmetrical if you allow for enough bootstrap samples. Many times, it is not used to get an estimate, but for getting the bootstrap percentile confidence interval. If the variable of interest is skewed in your sample, just report the median and MAD (median absolute deviation or interquartile range.