# Quantile absolute deviation of the Uniform distribution

**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.**

In this post, we derive the exact equation for the quantile absolute deviation around the median of the Uniform distribution.

## Preparation

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

\[\newcommand{\E}{\mathbb{E}} \newcommand{\PR}{\mathbb{P}} \newcommand{\Q}{\operatorname{Q}} \newcommand{\QAD}{\operatorname{QAD}} \newcommand{\median}{\operatorname{median}} \QAD(X, p) = \Q(|X - \median(X)|, p), \]

where \(\Q\) is a quantile estimator.

We are looking for the asymptotic value of \(\QAD(X, p)\). For simplification, we denote it by \(v_p\):

\[v_p = \lim_{n \to \infty} \E[\Q(|X-M|, p)], \]

where \(M\) is the true median of the distribution.

By the definition of quantiles, this can be rewritten as:

\[\PR(|X_1 - M| < v_p) = p, \]

which is the same as

\[\PR(-v_p < X_1 - M < v_p) = p. \]

Hence,

\[\PR(M - v_p < X_1 < M + v_p) = p. \]

If \(F\) is the CDF of the considered distribution, the above equality can be rewritten as

\[F(M + v_p) - F(M - v_p) = p. \tag{1} \]

## Uniform distribution

We consider the standard uniform distribution \(\mathcal{U}(0, 1)\) given by the CDF \(F(x)=x\) on \([0;1]\) with the median value \(M=0.5\). From(1), we have:

\[(0.5 + v_p) - (0.5 - v_p) = p, \]

which is the same as

\[v_p = p / 2. \]

Thus, if \(X \sim \mathcal{U}(0, 1)\),

\[\lim_{n \to \infty} \E[\QAD(X, p)] = \frac{p}{2}. \]

Here is the corresponding plot: