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.

  • $\begingroup$ 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 $\endgroup$ – Tim Feb 21 '18 at 10:09
  • $\begingroup$ @Tim I've added the example from the book. $\endgroup$ – An old man in the sea. Feb 21 '18 at 10:23
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    $\begingroup$ I don't get what do you mean by the example and what does your symbols mean. $\endgroup$ – Tim Feb 21 '18 at 10:26
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    $\begingroup$ I presume that the OP means the likelihood is unbounded for this mixture example. $\endgroup$ – Xi'an Feb 21 '18 at 10:44
  • $\begingroup$ @Tim, I hope now it's clearer. Sorry for the inconvenience. $\endgroup$ – An old man in the sea. Feb 21 '18 at 10:50

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