On The Density Estimation by Super-Parametric Method

TL;DR

This paper discusses the implementation and effectiveness of super-parametric density estimators, focusing on algorithm design, convergence, and numerical results, demonstrating their potential to unify nonparametric and parametric methods in statistics.

Contribution

It provides a detailed implementation and analysis of super-parametric density estimation algorithms, including strategies for window function selection and convergence, advancing the practical application of these estimators.

Findings

Algorithms converge rapidly and are mathematically effective.

Numerical results support the potential of super-parametric estimators.

The approach introduces a new paradigm in statistical density estimation.

Abstract

The super-parametric density estimators and its related algorism were suggested by Y. -S. Tsai et al [7]. The number of parameters is unlimited in the super- parametric estimators and it is a general theory in sense of unifying or connecting nonparametric and parametric estimators. Before applying to numerical examples, we can not give any comment of the estimators. In this paper, we will focus on the implementation, the computer programming, of the algorism and strategies of choosing window functions. B-splines, Bezier splines and covering windows are studied as well. According to the criterion of the convergence conditions for Parzen window, the number of the window functions shall be, roughly, proportional to the number of samples and so is the number of the variables. Since the algorism is designed for solving the optimization of likelihood function, there will be a set of nonlinear equations with a large number of variables. The results show that algorism suggested by Y. -S. Tsai is very powerful and effective in the sense of mathematics, that is, the iteration procedures converge and the rates of convergence are very fast. Also, the numerical results of different window functions show that the approach of super-parametric density estimators has ushered a new era of statistics.