  We introduce a new approach to deciding the number of clusters. The approach
is applied to Optimally Tuned Robust Improper Maximum Likelihood Estimation
(OTRIMLE; Coretto and Hennig 2016) of a Gaussian mixture model allowing for
observations to be classified as "noise", but it can be applied to other
clustering methods as well. The quality of a clustering is assessed by a
statistic $Q$ that measures how close the within-cluster distributions are to
elliptical unimodal distributions that have the only mode in the mean. This
nonparametric measure allows for non-Gaussian clusters as long as they have a
good quality according to $Q$. The simplicity of a model is assessed by a
measure $S$ that prefers a smaller number of clusters unless additional
clusters can reduce the estimated noise proportion substantially. The simplest
model is then chosen that is adequate for the data in the sense that its
observed value of $Q$ is not significantly larger than what is expected for
data truly generated from the fitted model, as can be assessed by parametric
bootstrap. The approach is compared with model-based clustering using the
Bayesian Information Criterion (BIC) and the Integrated Complete Likelihood
(ICL) in a simulation study and on two datasets of scientific interest.
Keywords: parametric bootstrap; noise component; unimodality; model-based
clustering
