Suppose I bootstrap the distribution of the sample mean. Normally, one would use the mean of the bootstrapped distribution as point estimate of the parameter and the s.d. as its standard error. The mean of the bootstrapped distribution is asymptotically equal to the sample estimate (i.e. for a large number of iterated draws).
Now suppose the mean, or more generally, parameter, has an asymetrical bootstrap distribution, so that the sample estimate of the parameter and the mean of the bootstrap distribution are likely to be unequal for a moderate amount of iterated draws. Should I still expect both to be asympotically equivalent, so as I increase the number of draws, both will be equal? And if this is so, is it customary practice to increase the number of iterations until they are equal before I report statistics?
In my practical case, both deviate after 1000 iterated draws. So I am unsure whether I should report the sample estimate or the mean of the bootstrap distribution of the parameter.