How to fit a mixture of a normal and a half normal distribution?

Question

I tried Expectation-Maximization (EM) based fitting using the mixfit function from the mixR package in the R environment. It yielded a normal mixture model with 2 components: 1) pi 0.21, mu: 0.47, sd: 0.31 2) pi 0.78, mu 2.55, sd: 1.16

However the fit would probably be better if the first component was fitted with half-normal (or truncated normal) as mixture density type instead of normal (with mu close to 0, no values below 0). The function mixfit can only deal with normal, gamma or beta distributions.

My question: is there a function suitable for EM based fitting of a mixture of a half normal (truncated normal) and a normal distribution?

Answer 1

E-step: write complete log-likelihood $$\sum_{i=1}^n \mathbb{I}_1(z_i) \log \pi_1 f(x_i;\mu_1,\sigma_1) + \mathbb{I}_2(z_i) \log \pi_2 f(x_i;\mu_2,\sigma_2)$$and take expectation $$\mathbb{E}_{\theta^0}\left[\sum_{i=1}^n \mathbb{I}_1(z_i) \log \pi_1 f(x_i;\mu_1,\sigma_1) + \mathbb{I}_2(z_i) \log \pi_2 f(x_i;\mu_2,\sigma_2)|x_1,\ldots,x_n\right]$$ as$$\sum_{i=1}^n \mathbb{P}(Z_i=1|x_i,\theta^0) \log \pi_1 f(x_i;\mu_1,\sigma_1) + \mathbb{P}(Z_i=2|x_i,\theta^0) \log \pi_2 f(x_i;\mu_2,\sigma_2)$$where $$\mathbb{P}(Z_i=1|x_i,\theta^0)=\pi_1^0 f(x_i;\mu_1^0,\sigma_1^0)\Big/\pi_1^0 f(x_i;\mu_1^0,\sigma_1^0)+\pi_2^0 f(x_i;\mu_2^0,\sigma_2^0)$$

M-step: maximise above in $\theta=(\pi_1,\mu_1,\sigma_1,\mu_2,\sigma_2)$, for instance $$\hat\pi_1=\frac{1}{n} \sum_{i=1}^n \mathbb{P}(Z_i=1|x_i,\theta^0)$$ Since$$f(x;\mu,\sigma)=\exp\{-(x-\mu)^2/2\sigma^2\}\big/\sqrt{2\pi}\sigma\Phi(-\mu/\sigma)$$the maximisation in $(\mu,\sigma)$ requires numerical optmisation tools.