  Generalized $k$-means can be incorporated with any similarity or
dissimilarity measure for clustering. By choosing the dissimilarity measure as
the well known likelihood ratio or $F$-statistic, this work proposes a method
based on generalized $k$-means to group statistical models. Given the number of
clusters $k$, the method is established under hypothesis tests between
statistical models. If $k$ is unknown, then the method can be combined with GIC
to automatically select the best $k$ for clustering. The article investigates
both AIC and BIC as the special cases. Theoretical and simulation results show
that the number of clusters can be identified by BIC but not AIC. The resulting
method for GLMs is used to group the state-level time series patterns for the
outbreak of COVID-19 in the United States. A further study shows that the
statistical models between the clusters are significantly different from each
other. This study confirms the result given by the proposed method based on
generalized $k$-means.
