# Does it make sense to calculate MLE for multimodal distributions?

The simplest examples of multimodal distributions I've seen are mixtures, namely mixtures of normals.

However, in this case, the Maximum Likelihood Estimator [MLE] doesn't make much sense. An example is described in the DeGroot and Schervish book.

The MLE for a mixture of a Normal$(0,1)$ and Normal$(\mu, \sigma^2)$ (both parameters unknown) for a sample $(x_1,...,x_n)$ are non-unique: $\hat \mu =x_k$ for $k \in \{1,...,n\}$ and, if we allow it, $\hat \sigma^2=0$.

Are there other examples of multimodal distributions where computing the MLE does make sense?

Any help would be appreciated.

• What exactly doesn't make sense..? You can use algorithms like EM to estimate the ML for a mixture distribution, to give an example. Search yourself for examples: google.com/search?q=maximum+likelihood+mixture – Tim Feb 21 '18 at 10:09
• @Tim I've added the example from the book. – An old man in the sea. Feb 21 '18 at 10:23
• I don't get what do you mean by the example and what does your symbols mean. – Tim Feb 21 '18 at 10:26
• I presume that the OP means the likelihood is unbounded for this mixture example. – Xi'an Feb 21 '18 at 10:44
• @Tim, I hope now it's clearer. Sorry for the inconvenience. – An old man in the sea. Feb 21 '18 at 10:50