From statistical theory, we often obtain results such as
$\sqrt n (\theta - \hat \theta) \rightarrow_d N(0, \sigma)$
ie we have a normal limiting distribution.
Because this formula says nothing about how quickly (with respect to $n$) this limiting behavior kicks in, we may do simulations to look at the finite sample behavior.
My current conundrum involves trying to define how one might measure and summarize this. The problem I'm working on is very similar to the case of logistic regression. Many readers will be familiar with the limiting distribution of the coefficients of logistic regression. However, consider examining the finite sample behavior of regression coefficients for logistic regression. A knee-jerk reaction (at least, my knee-jerk reaction) is to examine the finite sample mean and standard deviation and compare with the asymptotic behavior. Theoretically, the mean and standard deviation for any finite sample is non-finite. This is because for logistics regression, there is a positive probability (albeit quite small for most cases) that the data will be perfectly separable. If that's the case, the regression coefficients are non-finite, so both the mean and standard deviation are theoretically undefined. And it's not just a theoretic problem: if your simulation has a relatively small sample size and you simulate lots of example, one of them is likely to end up with non-finite estimates. So you while the limiting distribution has mean $\beta$ (i.e. unbiased), for any finite sample, the mean is undefined. That's not a helpful comparison to make.
So how would one summarize the finite sample behavior to compare with the asymptotic results in a standard manner for publication? My first thought is to use something like comparing the mean and standard deviation from the truncated distribution of the regression parameters (i.e. dropping the largest and smallest 1% of regression estimates and then fitting as a truncated normal, for example), which will be finite as long as the probability of perfect separation is less than the truncation level. But I'm curious if there's a standard method or metric or at least a paper that does this so I don't need to defend this method in my paper.
