Concavity in Data Analysis

Concave functions play a fundamental role in the structure of and minimization of badness-of-fit functions used in data analysis when extreme values of either parameters or data need to be penalized. This paper summarizes my findings about this role. I also describe three examples where concave functions are useful: building measures of badness-of-fit, building robust M -estimators, and building clustering objective functions. Finally, using the common thread of concavity, all three will be combined to build a comprehensive, flexible procedure for robust cluster analysis.