I'm running a nonparametric bootstrap to estimate a confidence interval for the mean of a sample. My question is during this loop is it correct (or acceptable) to compute the skewness and kurtosis on the randomly sampled data for each iteration? This way I have a distribution of kurtosis and skewness (would plotting a histogram for these 2 be of useful information?).
However, if I still want a single estimate for the kurtosis and skewness, this would just be computed on the distribution of bootstrapped means?
(i) a distribution can have the same skewness and kurtosis as the normal distribution but be very distinctly non-normal,
(ii) the variance of the distributions of sample skewness and kurtosis themselves relate to much higher moments; for example if the eighth moment of the population distribution is not finite, the variance of the distribution of the sample kurtosis is not finite.
If you want a single estimate of the skewness and kurtosis of the distribution from which the original sample was drawn, is there a reason you wouldn't simply take the sample skewness and kurtosis?
(Perhaps you might consider a bias adjustment of that estimate - as may be used with bootstrapping of other estimates, though.)
Note that the skewness and kurtosis of the resampled means will be different from that of the population the sample itself was drawn from.
Under the case that you actually want the skewness and kurtosis of the sampling distribution of the mean, then where you say "is it correct (or acceptable) to compute the skewness and kurtosis on the randomly sampled data for each iteration?" ... then that won't give you what you seek (because, again, the skewness and kurtosis of the original variable is different from the skewness and kurtosis of the sampling distribution of the mean).
If the original sample is not drawn from a normal distribution the distribution of the mean is not normal (though it will generally be closer to normal than the parent distribution). If you are interested in the shape of the distribution of the mean, while skewness and kurtosis might be useful, it may be better to consider what it is that knowing it's near to normal might be useful for ... and they may well tell you what aspects of the distribution of the mean to focus on.