Local-moment nonparametric density estimation of pre-binned data

Data-driven research is often hampered by privacy restrictions in the form of limited data sets or graphical representations without the benefit of raw data. Nonparametric techniques that circumvent these issues by using local moment information, thereby extending the piecewise-polynomial histograms, are developed. These methods utilize not only binned data counts, but also their conditional moments in order to better estimate the underlying density of the data. A particular polynomial spline density estimator can be found via penalized least-squares optimization with a roughness penalty. Two issues exist in regards to the original algorithm: (1) local moments for empty bins are undefined and (2) all moment information is treated equally, despite the fact that lower-order moments are more accurate than higher-order moments. Solutions to both issues are provided by using unconditional moments and incorporating a weight matrix into the optimization problem.