GaussianMixtureTest

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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].

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 Kasahara-Shimotsu test.

See also the usage by running

?gmm

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)

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.

• [1] Chen, J. & Li, P., 2009. Hypothesis Test for Normal Mixture Models: The EM Approach. the Annals of Statistics, 37(5 A), pp.2523–2542.

• [2] Chen, J., Tan, X. & Zhang, R., 2008. Inference for Normal Mixtures in Mean and Variance. Statistica Sinica, 18, pp.443–465.

• [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.1–33.