# Can anyone tell me what is this distribution?

**URL:** <https://discourse.julialang.org/t/can-anyone-tell-me-what-is-this-distribution/53726>\
**Category:** Statistics\
**Tags:** question, math\
**Created:** [January 21, 2021, 2:14pm UTC](https://discourse.julialang.org/t/can-anyone-tell-me-what-is-this-distribution/53726 "2021-01-21T14:14:15Z")\
**Posts on this page:** 1\
**Page:** 1

<div class="post-metadata">

**Author:** ![abulak](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/abulak/32/28314_2.png) [@abulak](https://discourse.julialang.org/u/abulak)\
**Post date:** [January 21, 2021, 2:14pm UTC](https://discourse.julialang.org/t/can-anyone-tell-me-what-is-this-distribution/53726/1 "2021-01-21T14:14:15Z")

</div>

In my hobby project I arrived at the following problem: I’m given a family of random variables: `Wₖ` for which I need to estimate cdf:

 ![W_histograms](https://global.discourse-cdn.com/julialang/original/3X/1/1/1119c9be09f9ba2624eb37afc60d4f520bb868ae.png)  
(the above is histogram approximation of pdf from 10^7 samples)

From theoretical considerations I know that

cdf(W₃)(w) = \frac{6}{\pi} \left(\sin^{-1}{\sqrt{w}} - \sin^{-1}{\sqrt{\frac{3}{4}}} \right)

It is clear though that for `k=4, 5, 6...` the pdfs are piecewise functions with `k=4` consisting of `4` functions: linear in `(0.63, 0.73)`, then exponential, another exponential(?), and linear close to `w=1`.  
Similarly for `k=5` there are four pieces, but as `k` grows the density becomes smoother (the pieces are visible up to `k=7`, or `8`, especially if you plot histograms of `sqrt(W)`.

Any hints from anybody who has a grasp of statistics would be much appreciated (I’m not doing statistics at all 😉

### Where does it come from?

W is the Shapiro-Wilk W-statistic, the one from SW-normality test, which I’m trying to compute/estimate more… rigorously.  
see [https://github.com/JuliaStats/HypothesisTests.jl/pull/124](https://github.com/JuliaStats/HypothesisTests.jl/pull/124)

### What have I tried

For `k >= 9` I found that there is a normalizing power transform:

W \mapsto (1-W\_k)^t \sim \mathcal{N}(\mu\_k, \sigma\_k)

This seems ok, but as `k` grows mse of this approximation (as computed by fitting `0.005:0.005:0.995` quantiles) stabilizes at around `0.004` with noticeably thicker tails and deficit around the mean.

starting from `k >= 28` a log transform

W \mapsto log(1 - W\_k) \sim \mathcal{N}(\mu\_k, \sigma\_k)

does a better job (it is visibly skewed for `k<=20`).

 ![normal_transforms_fit](https://global.discourse-cdn.com/julialang/original/3X/8/5/855a7e6bad77ff6c509524ccadd7c9a8c2c3a531.png)