 # Optimal quantile absolute deviation

Update: this blog post is a part of research that aimed to build a new measure of statistical dispersion called quantile absolute deviation. A preprint with final results is available on arXiv: arXiv:2208.13459 [stat.ME]. Some information in this blog post can be obsolete: please, use the preprint as the primary reference.

We consider the quantile absolute deviation around the median defined as follows:

$\newcommand{\E}{\mathbb{E}} \newcommand{\PR}{\mathbb{P}} \newcommand{\Q}{\operatorname{Q}} \newcommand{\OQAD}{\operatorname{OQAD}} \newcommand{\QAD}{\operatorname{QAD}} \newcommand{\median}{\operatorname{median}} \newcommand{\Exp}{\operatorname{Exp}} \newcommand{\SD}{\operatorname{SD}} \newcommand{\V}{\mathbb{V}} \QAD(X, p) = K_p \Q(|X - \median(X)|, p),$

where $$\Q$$ is a quantile estimator, and $$K_p$$ is a scale constant which we use to make $$\QAD(X, p)$$ an asymptotically consistent estimator for the standard deviation under the normal distribution.

In this post, we get the exact values of the $$K_p$$ values, derive the corresponding equation for the asymptotic Gaussian efficiency of $$\QAD(X, p)$$, and find the point in which $$\QAD(X, p)$$ achieves the highest Gaussian efficiency.

### Asymptotic consistency constants for QAD

Let us assume that $$X$$ follows the standard normal distribution $$\mathcal{N}(0, 1)$$. Since we want to achieve $$\lim_{n \to \infty} \E[\QAD(X, p)] = 1$$, we have

$\lim_{n \to \infty} \E[\QAD(X, p)] = \frac{1}{K_p}.$

Using the exact equation for the $$\QAD(X, p)$$ of the normal distribution, we get the exact value for the asymptotic consisency constant value:

$K_p = \dfrac{1}{\Phi^{-1}((p+1)/2)}.$

### Asymptotic Gaussian efficiency of QAD

In this section, we consider the asymptotic relative statistical efficiency of the $$\QAD$$ against the standard deviation under the normal distribution (Gaussian efficiency).

We have already derived the equation for the Gaussian efficiency in the non-scaled case. Using the value of $$K_p$$, it is easy to update the obtained equation for the scaled case:

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

Here is the corresponding plot:

We can see that the presented function is unimodal with a single maximum point. Let us denote the location of this point as $$\rho_o$$. This value can be obtained numerically:

$\rho_o \approx 0.861678977787423 \approx 86.17\%.$

### Optimal quantile absolute deviation

We define the optimal quantile absolute deviation by $$\OQAD(X) = QAD(X, \rho_o)$$. It can be interested to consider this measure of dispersion since it gives the highest Gaussian efficiency across all $$\QAD(X, p)$$ estimators ($$65.22\%$$). The corresponding breakdown point is $$1 - \rho_o \approx 13.83\%$$.