Shapiro-Wilk coefficients, computed exactly (up to `eps(Float64)`) using Arb with 512 bits of precision and itegration radius 32.0

fit_normal.jl

using Distributions
using Statistics
using LinearAlgebra
using Optim
using JSON

const sw_coeffs = let fname = "sw.json"
  d = JSON.parsefile(fname, dicttype=Dict{String, Vector{Float64}});
  sw_coeffs = Dict(parse(Int, k)=>v for (k,v) in d)
  @assert all(k->haskey(sw_coeffs, k), 5:189)
  sw_coeffs
end

function power_transform_residual(t, sample, qs, ideal_qs)

    x = quantile(sample, qs, sorted=true)
    x .= (one(eltype(x)) .- x).^t

    µ, σ = mean(x), std(x)

    x .= ( x .- µ ) ./ σ
    x .-= reverse(ideal_qs)

    return norm(x, 2)
end

function fit_quantiles(
    transform,
    empirical,
    ideal::Distribution,
    qs=0.005:0.005:0.995,
    ideal_qs=quantile.(ideal, qs)
)
    res = optimize(
        t->power_transform_residual(t, empirical, qs, ideal_qs),
        eps(Float64),
        1.0,
    )

    t = Optim.minimizer(res)
    norm_res = transform(t, empirical, qs, ideal_qs)

    µ, σ = let x = empirical
        x = (one(eltype(x)) .- x).^t
        mean(x), std(x)
    end

    @info "Obtained residual: $norm_res" t µ σ
    return t, µ, σ
end


@time results = map(5:60) do n
    @time begin
    qs=0.005:0.005:0.995
    normal_qs=quantile.(Normal(0,1), qs)
    W_sample = sort!(sample_Wstatistic(sw_coeffs[n], 1_000_000), rev=false)

    t, µ, σ = let qs=qs, normal_qs=normal_qs, sample=W_sample

        res = optimize(
            x->normal_power_fit_optim(first(x), sample, qs, normal_qs),
            eps(Float64),
            0.5,
            Brent(),
        )

        t = Optim.minimizer(res)
        norm_res, µ, σ = normal_power_fit(t, W_sample, qs, normal_qs)

        @info "For n=$n obtained" t=Optim.minimizer(res) µ σ norm_res
        @debug res
        t, µ, σ
    end
    end
    (t = t, µ = µ, σ = σ)
end

sw.json

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}

W_statistic.jl

using Statistics
using Random
using LinearAlgebra

function Wstatistic(X, sw_coeffs)
    @boundscheck @assert length(X) == length(sw_coeffs)
    @assert issorted(X)
    @assert last(X) - first(X) > length(X) * eps(eltype(X)) "Data seems to be constant!"

    µ = mean(X)
    S² = sum(x->abs2(x-µ), X)

    return dot(sw_coeffs, X)^2 / S²
end

function sample_Wstatistic(sw_coeffs::AbstractVector{T}, sample_size) where T
    Wstats = zeros(T, sample_size)
    tmp = similar(Wstats, length(sw_coeffs))
    for i in 1:sample_size
        randn!(tmp)
        sort!(tmp)
        Wstats[i] = Wstatistic(tmp, sw_coeffs)
    end
    return Wstats
end