Adds sphericity tests

Author: jiahao

src/StatisticalTests.jl

@@ -0,0 +1,26 @@
+using Distributions
+
+#Compute Mauchly's statistic (valid under assumption of multinormality)
+#Mauchly, 1940; Kendall and Stuart, 1968
+#n is the number of data points (samples)
+#This returns both the value of the test statistic and the expected distribution to test against
+function MauchlySphericityTestStatistic{T<:Real}(PopulationCovariance::AbstractMatrix{T}, SampleCovariance::AbstractMatrix{T}, n::Integer)
+    p = size(SampleCovariance)[1]
+    C = inv(PopulationCovariance) * SampleCovariance
+    l = (det(C)/(trace(C)/p)^p)
+    x = -n*log(l)
+    f = p*(p+1)/2-1
+    return x, Chisq(f)
+end
+
+#Johnstone's variant of the sphericity test
+#n, p>= 10 recommended
+#Johnstone (2001)
+function JohnstoneSphericityTestStatistic{T<:Real}(PopulationCovariance::AbstractMatrix{T}, SampleCovariance::AbstractMatrix{T}, n::Integer)
+    C = inv(PopulationCovariance) * SampleCovariance
+    v = max(eigvals(C))
+    mu=(sqrt(n-1)+sqrt(p))^2
+    sigma=sqrt(mu)*(1/sqrt(n-1)+1/sqrt(p))^(1/3)
+    (n*l-mu)/sigma #To be tested against Tracy-Widom with beta=1
+end
+