Bernstein Polynomial Model--Redefining Nonparametric Statistics and Nonparametric Models

Abstract

Traditionally statisticians thought that nonparametrically estimating quantities such as density function needs no model specification, i.e., a nonparametric density model simply means any nonnegative function f(x) with integral equal to one. However, in this case for any x with f(x)>0, the information for f(x) is zero (see Bickel, et.al. 1998). Ibragimov and Hasminskii (1982) also showed that no such nonparametric model even with some smoothness assumptions for which this information is positive. Therefore such ``nonparametric model'' is not useful and not even a model because it specifies almost nothing. Therefore, properly reducing the infinite dimensional parameter to a finite dimensional one is necessary. A working finite dimensional nonparametric density model is also necessary to apply the maximum likelihood method which, as well known, usually gives the most efficient estimate. Bernstein polynomial model (Guan, 2016, 2017) is the first such approximate finite dimensional nonparametric model. Using this model, we can obtain density estimate that enjoys an almost parametric rate of convergence and is much better than the traditional kernel density estimate.