# Mixture of Location-Scale Families

##### Posted on

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 $\sigma_2$