Simulations and modes of convergence - Biased and Inefficient

We often have theory that says $\sqrt{n} ({\hat{θ}}_{n} - θ) \overset{d}{\to} N (0, σ^{2}),$
and then do simulations to see how well the asymptotic approximation applies. After doing so, we often present tables of the empirical mean and standard deviation of ${\hat{θ}}_{n} .$ This doesn’t make a lot of sense.

Knowing that $\sqrt{n} ({\hat{θ}}_{n} - θ) \overset{d}{\to} N (0, σ^{2})$ doesn’t tell us anything about the moments of ${\hat{θ}}_{n}$ for any finite $n$ . Convergence in distribution does not imply convergence in mean. For example, ${\hat{θ}}_{n}$ could be maximum likelihood estimates in a logistic regression model. These have no finite moments for any finite $n$ , because they are infinite with positive probability.

However, the asymptotic result does tell us that the quantiles of ${\hat{θ}}_{n}$ , suitably scaled, should converge to those of the approximating Normal distribution. The median of ${\hat{θ}}_{n}$ should converge to $θ$ ; the MAD of ${\hat{θ}}_{n}$ (after scaling by $\sqrt{n}$ ) should converge to $σ$ , and the probability that $({\hat{θ}}_{n} - 1.96 \times \hat{s e} [{\hat{θ}}_{n}], {\hat{θ}}_{n} + 1.96 \times \hat{s e} [{\hat{θ}}_{n}])$ includes $θ$ should converge to 95%.

Knowing how good the asymptotic approximation is for these quantile-based statistics is usually sufficient to tell us if the approximation is useful in practice. We usually don’t care about the mean of ${\hat{θ}}_{n}$ in any substantive way, since if we did, we’d be more worried when it didn’t have one.

I think the usefulness of ‘robust statistics’ gets oversold a lot, but this really is a case where it’s meaningful and true to say that the median of an approximately-Normal variable is a better summary of the location parameter than the mean is. We should be presenting robust (ie, weakly continuous) summaries of the results of simulations motivated by asymptotics unless there’s a specific reason to care about moments.