[RFC]: skew-normal distribution

[quinn-dougherty]

Description

This RFC proposes the skew-normal distribution

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Checklist

• I have read and understood the Code of Conduct.
• Searched for existing issues and pull requests.
• The issue name begins with RFC:.

[github-actions]

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[kgryte]

Hey, @quinn-dougherty! Thanks for filing this RFC.

This would definitely be a nice distribution to have. Does entail, however, a more involved implementation. Namely, we'd need integration facilities (see SciPy) and Owen's T function (see https://www.jstatsoft.org/article/view/v005i05 (GPL)).

@Planeshifter May have a better sense of how difficult this distribution may be to implement.

[jqhoogland]

For now, this is getting the job done for me:

skew_normal(params) = {
    loc = params.loc
    scale = params.scale
    shape = params.shape

    {|x| (2 / scale) * pdf(normal(loc, scale), x) * cdf(normal(loc, scale), shape * x)}
}

(when a function suffices)

Or this incredibly janky thing:

skew_normal(params) = {
    _skew_normal(params) = {
        loc = params.loc
        scale = params.scale
        shape = params.shape

        {|x| (2 / scale) * pdf(normal(loc, scale), x) * cdf(normal(loc, scale), shape * x)}
    }

    range = List.upTo((params.loc - params.scale * 3) * 100, (params.loc + params.scale * 3) * 100)
    PointSet.makeContinuous(range |> map({|x| {x: x/ 100, y: _skew_normal(params)(x/100)}}))
}

(when I need a proper distribution)

[Prog-Jacob]

Hey, @kgryte!

I would like to work on that. I checked the SciPy source code you shared; they are using direct implementation of the formula here. On the contrary, in the paper, they are utilizing Owen's T function properties to bring down the $\text{shape } (a)$ to $0 \le a \le1$. Then, they use one of six approximation methods depending on the value of h and a. One of these methods is the Gaussian quadrature, which I think is sufficient for now and wouldn't require integration facilities. Also, Owen's T function will be needed in the Multivariate Normal Distributions (GSoC listed ideas), and this can be a good starter for whom will work on it.

[kgryte]

@Prog-Jacob As a start, feel free to submit an initial PR which adds the pdf to @stdlib/stats/base/dists/skew-normal/pdf.

[kgryte]

@Prog-Jacob Or perhaps take an initial pass at adding Owen's T to @stdlib/math/base/special/owens-t, based on Boost: https://github.com/boostorg/math/blob/15c40fae1610fd83728d75e96d7377fe55f976f7/include/boost/math/special_functions/owens_t.hpp#L1030

[Prog-Jacob]

@kgryte Boost is implementing the paper you referenced – Fast and Accurate Calculation of Owen's T Function. It looks hefty; should I initially focus on supporting Skewed Normal Distributions and work on Owen's T function later on? what do you think?

[kgryte]

@Prog-Jacob Sure. Sounds good.