Asymptotic Gaussian efficiency of the quantile absolute deviation
The below text contains an intermediate snapshot of the research and is preserved for historical purposes.
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 (
where
Let us consider
If
where
The asymptotic variance of the sample quantile estimator for distribution with probability density function
Using
The asymptotic variance of the standard deviation estimator is well-known:
Finally, we are ready to draw the equation for the Gaussian efficiency of