This package aim is to implement in Julia some robust mean estimators (one-dimensional for now). See Mean Estimation and Regression Under Heavy-Tailed Distributions: A Survey or The Robust Randomized Quasi Monte Carlo method, applications to integrating singular functions for recent surveys.
I am currently trying some stuff on the package about "robust moving average"
Computing the empirical mean over a data set is one of the most common operations in data analysis.
However, this operation is not robust to outliers or contaminated samples.
Robust mean estimators are mean estimator that are robust (in some sense) against such outliers or contaminated samples.
using Distributions
using RobustMeansn = 8 * 7
M = 10^5 # M = 10^7 is used for the plot
α = 3.1
distribution = Pareto(α)
μ = mean(distribution) # True mean
σ = std(distribution) # True std
x = rand(distribution, M, n) # M realization of samples of size n# Store all the realizations into a Dictionary
p = 1 # Parameter of MinskerNdaoud
δ = 3exp(-8) # 0.001
estimators = [EmpiricalMean(), Catoni(σ), Huber(σ), LeeValiant(), MinskerNdaoud(p)]
short_names = ["EM", "CA", "HU", "LV", "MN"]
estimates = Dict{MeanEstimator,Vector}()
for estimator in estimators
estimates[estimator] = [mean(r, δ, estimator) for r in eachrow(x)]
endusing StatsPlots, LaTeXStrings
gr()
plot_font = "Computer Modern" # To have nice LaTeX font plots.
default(
fontfamily = plot_font,
linewidth = 2,
label = nothing,
grid = true,
framestyle = :default
)
# The plot
begin
plot(thickness_scaling = 2, size = (1000, 600))
plot!(Normal(), label = L"\mathcal{N}(0,1)", c = :black, alpha = 0.6)
for (ns, s) in enumerate(estimators)
W = √(n) * (estimates[s] .- μ) / σ
stephist!(W, alpha = 0.6, norm = :pdf, label = short_names[ns], c = ns)
vline!([quantile(W, 1-δ)], s = :dot, c = ns)
end
vline!([0], label = :none, c = :black, lw = 1, alpha = 0.9)
yaxis!(:log10, yminorticks = 9, minorgrid = :y, legend = :topright, minorgridlinewidth = 1.2)
ylims!((1/M*10, 2))
ylabel!(L"\sqrt{n}(\hat{\mu}_n-\mu)/\sigma", tickfonthalign = :center)
xlims!((-5, 10))
yticks!(10.0 .^ (-7:-0))
end