GaussianMixtureTest
This package implements the EM test for number of components of univariate Gaussian Mixture. The conventional log likelihood test can not be used to test the number of components because the Fisher regularity conditions are violated in Gaussian Mixture case [1].
This package follows the method of [3] but with no regression covariates. Note that the asymptotic distribution of the test statistic is that of the maximum of C0
dependent $Chi^2(2)$ random variables which has no closed form when the null distribution has more than 1 component. However the p value can be obtained via simulation.
In addition the typical EM algorithm may fail to give a consistent estimate of Gaussian Mixture parameters. This package still uses EM but add a penalty term on the log likelihood which ensures the consistency [2].
Usage
To install this package, run
Pkg.add("GaussianMixtureTest")
The major functions are gmm
, gmmrepeat
,asymptoticdistribution
and kstest
. gmm
estimates the parameters via EM algorithm. gmmrepeat
repeat gmm
for multiple starting values. asymptoticdistribution
simulates the asymptotic distribution of the test statistic when the number of components is greater than 2. kstest
conducts the KasaharaShimotsu test.
See also the usage by running
?gmm
Examples
See the example code also in runtests.jl
using Distributions
using GaussianMixtureTest
mu_true = [2.0858,1.4879]
wi_true = [0.0828,0.9172]
sigmas_true = [0.6735,0.2931]
m = MixtureModel(map((u, v) > Normal(u, v), mu_true, sigmas_true), wi_true)
x = rand(m, 1000);
asymptoticdistribution(x, wi_true, mu_true, sigmas_true, debuginfo=true);
#Estimate the parameters with two components
wi, mu, sigmas, ml = gmm(x, 2)
#Do the KS test for C=2 v.s. C=3
kstest(x, 2)
Acknowledgement
Thanks Dr. Shimotsu and Dr. Kasahara for nicely providing their original R code and their detailed explanations. Several implementation details of this package are borrowed from their R code.
Reference

[1] Chen, J. & Li, P., 2009. Hypothesis Test for Normal Mixture Models: The EM Approach. the Annals of Statistics, 37(5 A), pp.2523โ2542.

[2] Chen, J., Tan, X. & Zhang, R., 2008. Inference for Normal Mixtures in Mean and Variance. Statistica Sinica, 18, pp.443โ465.

[3] Kasahara, H. & Shimotsu, K., 2015. Testing the Number of Components in Normal Mixture Regression Models. Journal of the American Statistical Association (to appear), pp.1โ33.