Quantile absolute deviation of the Normal distribution
THIS POST IS OUTDATED. Up-to-date preprint: PDF / arXiv:2208.13459 [stat.ME]
The below text contains an intermediate snapshot of the research and is preserved for historical purposes.
The below text contains an intermediate snapshot of the research and is preserved for historical purposes.
In this post, we derive the exact equation for the quantile absolute deviation around the median of the Normal distribution.
Preparation
We consider the quantile absolute deviation around the median defined as follows:
where
We are looking for the asymptotic value of
where
By the definition of quantiles, this can be rewritten as:
which is the same as
Hence,
If
Normal distribution
We consider the standard normal distribution
Using
which is the same as
Thus, if
Here is the corresponding plot: