How to identify points and an unknown distribution in a two type clustering problem?

Question

I have a data set that contains two types of points. The first type of points come from an N(0,1) distribution. The second type of points come from an N(m,v) distribution for some real m and some positive, real v. The objective is to classify each point as type 1 or type 2, and to identify m & v. We have no apriori information about m & v. Any ideas?

Answer 1

You could use a mixture model to separate out the components. The data generating process can be represented as follows:

Let:

$z_i$: be the type (1 or 2) for the $i^{th}$ observation,

$y_i$ be the $i^{th}$ observation.

Then you have:

$f(y_i|z_i=1) \sim N(0,1)$

$f(y_i|z_i=2) \sim N(m,v)$

$P(z_i=1) = \pi$ and

$P(z_i=2) = 1-\pi$.

The likelihood function is given by:

$L(m,v,\pi|-) = \sum_{y_i} \pi f(y_i|z_i=1) + (1-\pi) f(y_i|z_i=2)$

You can then use either EM or MCMC to estimate for the model parameters.