Mixture Model Model Selection

Visser & Speekenbrink: Sec 2.6

Method 1: Likelihood Ratio Test

Likelihood ratio test can be used when regularity conditions and models have the same number of mixture components.

Alternatively, we can use bootstrapping method instead of the actual chi-square distribution to test $k$ components vs. $k + l$ components. The process are

1. Fit the $k, k+l$ component mixture models to obtain MLE estimates $\hat{\theta}_k, \hat{\theta}_{k+l}$ and compute the likelihood ratio $\Lambda_{obs}$

2. For $i=1,...,M$:

   1. Use the fitted $k$ component model to simulate a bootstrap sample

   2. Fit the $k, k+l$ component mixture models to the bootstrap data and calculate the likelihood ratio

3. Calculate the proportion of bootstrap likelihood ratio which are equal or larger than observed.

Method 2: Information Criteria

See information criteria