I am in the same boat so let's work on this together. From what I understand, there are global spatial statistics and local spatial statistics. The global ones, such as Moran's I or Getis-Ord Gi (without the *), capture global clustering and hotspots for the entire region of your dataset, whereas the local statistics such as Getis-Ord Gi* capture the local clustering and hotspots better.
Kernel density is another way of capturing density or space-use. Kernel density is non-parametric meaning it doesn't identify statistically significant hotspots.
Hi Cilla,
Thanks for your reply. I've been digging through some papers, and I largely came to the same conclusions you have. An excellent piece of citable research that helps explains some of the benefits of the Getis-Ord Gi* (and some of the negative aspects of the Moran's I), is Braithwaite and Li (2007), "Transnational Terrorism Hot Spots: Identification and Impact Evaluation." The discussion is largely on pages 285 - 287.
I wish ArcGIS were a little more clear on the pros and cons of each tool, especially given how more and more disciplines are adapting geospatial analyses for their research.