Asymptotic Properties of the MLE

Every worked example in the MLE lesson carried a quiet asterisk: $\hat\sigma^2_{\text{MLE}}$ is biased in finite samples, dividing by $n$ instead of $n-1$. If the MLE is not even unbiased, why is it the estimator every quant desk reaches for by default? The answer is asymptotic: as the sample size grows, the MLE's finite-sample flaws wash out, and three guarantees kick in simultaneously — it converges to the truth (consistency), its sampling distribution becomes exactly Normal at the $1/\sqrt n$ rate (asymptotic normality), and the variance of that limiting Normal distribution is the smallest one any estimator could hope to achieve (asymptotic efficiency — the Cramér-Rao bound, attained in the limit). This note proves all three, then extends the second via the delta method, which is what actually lets a risk system report a standard error for a derived quantity — a Sharpe ratio, an annualised volatility, a VaR multiplier — built from an MLE, rather than for the raw parameter alone.

Learning objectives. By the end of this note you should be able to: