# Some thoughts on finite sample properties of an estimator

I derived the mean squared errors (MSE) of a consistent estimator for two models: restricted and unrestricted. In addition, I showed that both has the same rate of convergence (that is, the asymptotic expressions for the MSE's are equal).

When I do Monte Carlo simulations I observe that the MSEs slowly become closer as the sample size increases. This was expected since both are asymptotically equal. But for $$n=500$$ (the sample size), they are still far from each other. Someone more experienced than me asked if it isn't suggesting that I commited errors on the theoretical derivations.

I'm thinking myself that if there were some big flaw in theory, both MSE shouldn't get closer with $$n$$. The fact that for a finite sample size, the MSEs are still far from each other means nothing for the asymptotic theory.

Context

I want to generate an estimator from the MSE of the unrestricted model. As the MSE of the restricted model is simpler, and both MSE's are asymptotically equivalent, it makes sense to consider the restricted MSE. As commented above, this strategy may result in poor finite sample performance (with asymptotic justification though).

My question is: do you see anything wrong with my ideas?

Thanks in advance!

## 1 Answer

It is true that asymptotic properties often appear for surprisingly small sample sizes (read Geyer on "non-N asymptotics" as to why that might be the case), but this does not mean that it will happen in all cases. So while the concern you received is legitimate in principle, the fact that larger sample sizes verify the equivalence, validate your theoretical results.

As regards using the one estimator in place of the other though, "simplicity" (of computation?) is not a good enough argument, except if you will work with sample sizes where the asymptotic equivalence does emerge. "Do the extra work" would be an anticipated response, except if you can show that this "extra work" is immense. "Asymptotic justification" is accepted when finite-sample results and formulas are not available, or when it would be too cumbersome and time-consuming to obtain them.

• Thank you for the comment. Yes, simplicity of computation. But another estimator based on the unrestricted MSE is also provided (which is much more time consuming). This showed good finite sample performance though. – Celine Harumi May 1 '20 at 13:53