Mixture of Location-Scale Families
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This note is for Chen, J., Li, P., & Liu, G. (2020). Homogeneity testing under finite location-scale mixtures. Canadian Journal of Statistics, 48(4), 670–684.
Testing problem for the order of finite mixture models
Ghosh & Sen (1985): the hard-to-manage asymptotic properties of the likelihood ratio test
the modified likelihood ratio test and the EM-test, lead to neat solutions for finite mixtures of univariate normal distributions, finite mixtures of single-parameter distributions, and several mixture-like models.
But the problem remains challenging, and still no generic solution for location-scale mixtures.
The article, provide an EM-test solution for homogeneity for finite mixtures of location-scale family distributions.
Let ${f(x;\theta):\theta\in \Theta}$ be a parametric distribution family. A finite mixture model of order $m$ is
\[f(x;G) = \sum_{j=1}^m \alpha_j f(x;\theta_j)\]with the mixing distribution $G(\theta)$ given by
\[G(\theta) = \sum_{j=1}^m\alpha_j 1(\theta_j\le \theta)\]Consider the location-scale family
\[f(x;\theta) = \frac 1\sigma f_0\left(\frac{x-\mu}{\sigma}\right)\]The log-likelihood function is given by
\[\ell_n(G) = \sum_{i=1}^n \log f(x_i;G)\]A well-known undesirable property of the location-scale mixture is that $\ell_n$ is unbounded.
Three ways to retain likelihood-based inference
- identify the local maximum point in the interior of the parameter space with the largest likelihood value among all interior local maximum points
- places constraints on $\sigma_1$ and $\sigma_2$
- penalize the log-likelihood with a penalty function on $\sigma_1$ and $\sigma_2$