Asymptotic Gaussian efficiency of the quantile absolute deviation

I have already discussed the concept of the quantile absolute deviation in several previous posts. In this post, we derive the equation for the relative statistical efficiency of the quantile absolute deviation against the standard deviation under the normal distribution (so call Gaussian efficiency).

In the context of this post, we consider the quantile absolute deviation ($\operatorname{QAD}$) around the median:

$$ \newcommand{MAD}{\operatorname{MAD}} \newcommand{QAD}{\operatorname{QAD}} \newcommand{Q}{\operatorname{Q}} \newcommand{SD}{\operatorname{SD}} \newcommand{QHF}{\operatorname{Q}_{\operatorname{HF7}}} \newcommand{QHD}{\operatorname{Q}_{\operatorname{HD}}} \newcommand{QTHD}{\operatorname{Q}_{\operatorname{THD-SQRT}}} \newcommand{exp}{\operatorname{exp}} \newcommand{erfinv}{\operatorname{erf}^{-1}} \newcommand{E}{\mathbb{E}} \newcommand{V}{\mathbb{V}} \QAD(X, p) = \Q(|X - \Q(X, 0.5)|, p), $$

where $Q$ is a sample quantile estimator, X is a sample of i.i.d. random variables $X = \{ X_1, X_2, \ldots, X_n \}$.

Let us consider $X$ from the standard normal distribution: $X \sim \mathcal{N}(0, 1)$. For the normal model, $\E[\Q(X, 0.5)] = 0$. Therefore,

$$ \lim_{n \to \infty} \QAD(X, p) = Q(|X|, p). $$

If $X$ follows the standard normal distribution, $|X|$ follows the standard half-normal distribution. The probability density function and the quantile function of the standard half-normal distribution are defined as follows:

$$ f_{\operatorname{HN}}(x) = \sqrt{\frac{2}{\pi}} \operatorname{exp}(-x^2/2), \quad Q_{\operatorname{HN}}(p) = \sqrt{2} \erfinv(p), $$

where $\erfinv$ is the inverse error function.

The asymptotic variance of the sample quantile estimator for distribution with probability density function $f$ and quantile function $Q$ is defined as follows:

$$ \lim_{n \to \infty} \V(Q_n(X, p)) = \frac{p(1-p)}{n f(Q(p))^2}. $$

Using $f_{\operatorname{HN}}$ and $Q_{\operatorname{HN}}$, we get

$$ \lim_{n \to \infty} \V(\QAD_n(X, p)) = \frac{\pi p(1-p)}{2n} \operatorname{exp}\Big(2\big(\erfinv(p) \big)^2 \Big). $$

The asymptotic variance of the standard deviation estimator is well-known:

$$ \lim_{n \to \infty} \V(\SD_n) = \frac{1}{2n}. $$

Finally, we are ready to draw the equation for the Gaussian efficiency of $\QAD$:

$$ \lim_{n \to \infty} e(\QAD_n(X, p),\; \SD_n(X)) = \lim_{n \to \infty} \frac{\V[\SD_n(X)]}{\V[\QAD_n(X, p)]} = \Bigg( \pi p(1-p) \exp\Big(2\big(\erfinv(p) \big)^2 \Big) \Bigg)^{-1}. $$