EpdTest

EpdTest.jl is a Julia library created for the paper Neyman’s C(α) test for the shape parameter of the exponential power class.

Installation

Through the pkg REPL mode by typing

] add "https://github.com/lukketotte/EpdTest.jl"

Recreating results

To recreate the second column of Figure 3

using Distributed
@everywhere using EpdTest, DataFrames

N, nsim = ([50, 100, 500], 10000);
p  = range(1., 4, length = 20);

simDat = DataFrame(n = repeat(N, inner = length(p)),
                   p = repeat(p, length(N)), value = 0.0)

# row 1
for i ∈ 1:length(N)
    β = pmap(kurt -> simSize(Epd(0.0, 1.0, kurt), N[i], nsim, 1), p)
    simDat[simDat.n .== N[i], :value] = β
end

# row 2
for i ∈ 1:length(N)
    β = pmap(kurt -> simSize(Epd(0.0, 1.0, kurt), N[i], nsim, "Normal"), p)
    simDat[simDat.n .== N[i], :value] = β
end

# row 3
for i ∈ 1:length(N)
    β = pmap(kurt -> simSize(Epd(0.0, 1.0, kurt), N[i], nsim, 3.), p)
    simDat[simDat.n .== N[i], :value] = β
end

To recreate Figure 5

using Distributed
@everywhere using EpdTest, DataFrames, Distributions

# MC adjusted sizes
αLapGel = [0.51, 0.235, 0.135, 0.083]
αLap = [0.0625, 0.057, 0.053, 0.051]

# Sample sizes and DF's of the t-distribution
N = [20, 50, 100, 200]
ν = [1, 2, 3, 4, 5, 6, 7]

# Based on the test outlined in Gel, 2010
simDat = DataFrame(n = repeat(N, inner = length(ν)), df = repeat(ν, length(N)), value = 0.0)
for i ∈ 1:length(N)
    β = pmap(df -> simSize(TDist(df), N[i], 50000, "Laplace",
        quantile(Chisq(1), 1-αLapGel[i])), ν)
    simDat[simDat.n .== N[i], :value] = β
end

# Based on the EPD test
simDat = DataFrame(n = repeat(N, inner = length(ν)), df = repeat(ν, length(N)), value = 0.0)
for i ∈ 1:length(N)
    β = pmap(df -> simSizeLaplace(TDist(df), N[i], 50000, 1.,
        quantile(Chisq(1), 1-αLap[i]), χ=true), ν)
    simDat[simDat.n .== N[i], :value] = β
end

The likelihood ratio test based on the empirical likelihood is not part of the package, but the test together with code for Figure 5 is

function empLik(x::AbstractVector{<:Real}, m::Integer)
    n = length(x)
    m <= √n || throw(DomainError(m, "m must be < √n"))
    sort!(x)
    p = 1
    for i ∈ 1:n
        p *= (2*m) / (n*(x[minimum([i+m, n])] - x[maximum([i-m, 1])]) *
            pdf(Laplace(median(x), mean(abs.(x .- median(x)))), x[i]))
    end
    p
end

function empLikTest(x::AbstractVector{<:Real})
    n = length(x)
    m = Integer(floor(√n))
    alt = zeros(m)
    for i ∈ 1:m
        alt[i] = empLik(x, i)
    end
    log(minimum(alt))
end

function simSizeEmpLap(d::D, n::N, nsim::N, critical::T) where
    {D <: ContinuousUnivariateDistribution, N <: Integer, T<: Real}
    sims = [0. for x in 1:nsim]
    for i in 1:nsim
        t = empLikTest(rand(d, n))
        sims[i] = abs(t) >= critical ? 1 : 0
    end
    mean(sims)
end

simDat = DataFrame(n = repeat(N, inner = length(ν)), df = repeat(ν, length(N)), value = 0.0)
crit = [7.662, 9.213, 10.478, 11.616]

for i in 1:length(N)
    β = pmap(df -> simSizeEmpLap(TDist(df), N[i], 50000, crit[i]), ν)
    simDat[simDat.n .== N[i], :value] = β
end

To recreate the applications for the bivariate normal case with 50 observations subsetted from the weather data. To recreate parts of the results in Dörr et. al. (2021), the subsample is selected through R using RCall in Julia.

# From RandomFields package in R
X = load("weather.csv") |> DataFrame
X = Matrix(X)

# Requires the RCall package
# gives the same indeces as Ref
idx = RCall.rcopy(R"""
RNGkind(sample.kind = "Rounding")
set.seed(0721)
idx = sample(1:157, 50)
""")

N = 10000
sim(n) = reshape(rand(MvNormal([0,0], diagm([1., 1.])), n), n, 2)

sims = [BivariateNormalTest(sim(50))^2 for i in 1:N]
mean(sims .> BivariateNormalTest(X[idx,:])^2)

sims = [JB(sim(50))^2 for i in 1:N]
mean(sims .> JB(X[idx,:])^2)

sims = [DEHU(sim(50)) for i in 1:N]
mean(sims .> DEHU(X[idx,:]))

References

• Dörr, P., Ebner, B. and Henze, N. A new test of multivariate normality by a double estimation in a characterizing pde. Metrika, 84(3):401-427, 2021.