# Comparing standard error asymptotics of standard deviation and mean absolute deviation estimators

I was reading Chapter 4 of Jean-Philippe Bouchaud's book "Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management" and in section 4.2.2 author was comparing standard errors of standard deviation and mean absolute deviation estimators. Given $$N$$ samples $$X_1, ..., X_N$$, let $$\sigma_e^2$$ be an estimator of the variance, i.e. $$\sigma_e^2 = \frac{1}{N} \sum_{i=1}^N (X_i - m_e)^2$$ where $$m_e$$ is a mean estimator, i.e. $$m_e = \frac{1}{N} \sum_{i=1}^N X_i$$.

Then the author procedes (where I think he made a typo) to approximate (when $$N$$ is large) the standard error of $$\sigma_e^2$$ as follows:

\begin{aligned} [\Delta (\sigma_e^2)] [\Delta (\sigma_e^2)]^2 &= \text{Var}\left(\frac{1}{N} \sum_{i=1}^N (X_i - m_e)^2\right) \\ &\approx \text{Var}\left(\frac{1}{N} \sum_{i=1}^N (X_i - m)^2\right) \\ &= \frac{\langle (X_1-m)^4 \rangle - \langle (X_1-m)^2 \rangle^2}{N} \\ &= \frac{\sigma^4}{N}(2 + \kappa) \end{aligned}

Then comes the confusing part: he then says that $$\frac{\Delta \sigma_e}{\sigma} = \frac{1}{2 \sqrt{N}}\sqrt{2 + \kappa} \approx \frac{1}{\sqrt{2N}} \left(1 + \frac{1}{4}\kappa\right)$$ Although I agree with the second approximation when $$\kappa$$ is small, why does the first equation hold? We do have $$\frac{\Delta\sigma_e^2}{\sigma^2} = \frac{\sqrt{2 + \kappa}}{\sqrt{N}}$$ from above, but where does an extra $$\frac{1}{2}$$ come from when you look at $$\Delta{\sigma_e}$$ instead of $$\Delta{\sigma_e^2}$$? Is it simply a typo or am I overlooking an obvious approximation?

Remark: if you also found the notation here confusing, from my understanding, $$\Delta \sigma_e^2$$ is standard error for the ("typical") estimator of $$\sigma^2$$, which is $$\sqrt{Var(\frac{1}{N} \sum_i (X_i - m_e)^2)}$$, while $$\Delta \sigma_e$$ here is standard error for the estimator of $$\sigma$$, which is $$\sqrt{Var(\sqrt{\frac{1}{N} \sum_i (X_i - m_e)^2)}}$$.

• Any idea why there is a subscript $_e$ (when working with $X$)? Mar 22 at 11:39
• subscript e is used for estimators Mar 22 at 12:09