Researchers working with spatial statistics techniques may be satisfied with the tools available in the desktop GIS packages. ESRI statistics tools will largely be found in the Spatial Statistics Toolbox and the Geostatistical Analyst Extension. Processes from both are available as toolboxes tools and ArcPy funcions. ArcGIS Pro also has a suite of interpolation tools available through the Geostatistcal Wizard in the analysis ribbon. QGIS users will find many applicable tools built in to the processing toolbox, in particular the SAGA GIS Geostatistical and Raster Creation tools although there are other options.
Quantitative scholars whose workflows are not already tied into standard GIS applications may be better served through other platforms. In particular R has a wealth of useful libraries, see the R page of this guide as well as the excellent CRAN Task View: Analysis of Spatial Data.
The Center for Spatial Data Science at the University of Chicago is a great resource for training materials on methods. They also provide a number of useful tools including GeoDa a free and easy to use software tool for exploratory spatial data analysis, including spatial autocorrelation statistics and spatial regression analysis. They also provide PySAL, the Python Spatial Analysis library provides of tools for spatial data analysis including cluster analysis, spatial regression, spatial econometrics as well as exploratory analysis and visualization.