speed up Semicirle sampling (#1580)

* speed up Semicirle sampling

* fix rand for Semicircle

* add semicircle rand tests

* Update src/univariate/continuous/semicircle.jl

Co-authored-by: David Widmann <devmotion@users.noreply.github.com>

* improve semicirle rand tests

Co-authored-by: David Widmann <devmotion@users.noreply.github.com>

Author: jw3126

src/univariate/continuous/semicircle.jl

@@ -72,4 +72,14 @@ function cdf(d::Semicircle, x::Real)
     end
 end
 
+function rand(rng::AbstractRNG, d::Semicircle)
+    # Idea:
+    # sample polar coodinates r,θ 
+    # of point uniformly distributed on radius d.r half disk
+    # project onto x axis
+    θ = rand(rng) # multiple of π
+    r = d.r * sqrt(rand(rng))
+    return cospi(θ) * r
+end
+
 @quantile_newton Semicircle

test/semicircle.jl

@@ -1,5 +1,6 @@
 using Distributions
-using  Test
+using Random: MersenneTwister
+using Test
 
 d = Semicircle(2.0)
 
@@ -37,3 +38,31 @@ d = Semicircle(2.0)
 @test quantile(d,  .0) == -2.0
 @test quantile(d,  .5) ==   .0
 @test quantile(d, 1.0) == +2.0
+
+rng = MersenneTwister(0)
+for r in rand(rng, Uniform(0,10), 5)
+    N = 10^4
+    semi = Semicircle(r)
+    sample = rand(rng, semi, N)
+    mi, ma = extrema(sample)
+    @test -r <= mi < ma <= r
+
+    # test order statistic of sample min is sane
+    d_min = Beta(1, N)
+    lo = quantile(d_min, 0.01)
+    hi = quantile(d_min, 0.99)
+    @test lo < cdf(semi, mi) < hi
+
+    # test order statistic of sample max is sane
+    d_max = Beta(N, 1)
+    lo = quantile(d_max, 0.01)
+    hi = quantile(d_max, 0.99)
+    @test lo < cdf(semi, ma) < hi
+
+    # central limit theorem
+    dmean = Normal(mean(semi), std(semi)/√(N))
+    @test quantile(dmean, 0.01) < mean(sample) < quantile(dmean, 0.99)
+
+    pvalue = pvalue_kolmogorovsmirnoff(sample, semi)
+    @test pvalue > 1e-2
+end