On The Density Estimation by Super-Parametric Method

The super-parametric density estimator and its related algorism are suggested by Y. -S. Tsai et al [7]. The number of parameters is unlimited in the super- parametric estimator and it is a general theory in sense of unifying or connecting nonparametric and parametric estimator. Before applying to numerical examples, we can not give any comment of the estimator. 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. 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 are many non-linear equations with many variables. The results show that algorism suggested by 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. From the numerical results of different window functions, we find that the approach of super-parametric density estimator has ushered a new era of statistic.