1

How could I fit data with observations from one Dirac delta component and $n$ normal distributed components? Where $n$ usually is between 1 and 5.

My prior knowledge is that one component really is a Dirac delta distribution. I know its location in an $\mathbb{R}^2$ space.

I don't know the precise mean and not at all the variance of the normal distributed components.

Now I would like to find mean and variance of the normal distributed components and classify my data to be able to weight the components against each other.

Can you recommend any R packages for this? I also considered julia's touring package or stan. As you see I am a novice in the realm and would be happy to devour any advice.

Jona Engel
  • 111
  • 3

1 Answers1

1

Assume that $X_i \sim \pi \delta_{x_0} + (1-\pi) N(\mu, \Sigma)$. If you know $x_0$, then, the natural estimate of $\pi$ is $$ \hat \pi = \frac1n \sum_{i=1}^n 1\{X_i = x_0\}. $$ You then fit a normal distribution to the rest of the data $\{X_i:\;X_i \neq x_0\}$ to get an estimate of $\mu$ and $\Sigma$.

passerby51
  • 1,573
  • 8
  • 11
  • Thanks! The formal part is clear to me. How would you suggest to implement this? Ideally in and _elegant_ and _simple_ way? I was first looking at `R`'s `mclust` and realized it can only handle normal distribution. – Jona Engel Jul 05 '21 at 14:36