Nonparametric estimation of cumulative distribution function and probability density function of continuous random variables is a basic and central problem in probability theory and statistics. Although many methods such as kernel density estimation have been presented, it is still quite a challenging problem to be addressed to researchers. In this paper, we proposed a new method of spline regression, in which the spline function could consist of totally different types of functions for each segment with the result of Monte Carlo simulation. Based on the new spline regression, a new method to estimate the distribution and density function was provided, which showed significant advantages over the existing methods in the numerical experiments. Finally, the density function estimation of high dimensional random variables was discussed. It has shown the potential to apply the method in classification and regression models.

中文翻译：

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概率分布估计及其在贝叶斯分类和最大似然回归中的应用。

连续随机变量的累积分布函数和概率密度函数的非参数估计是概率论和统计学中的基本和中心问题。尽管已经提出了诸如核密度估计之类的许多方法，但是仍然要解决给研究人员一个相当具有挑战性的问题。在本文中，我们提出了一种样条回归的新方法，其中，通过蒙特卡罗模拟的结果，样条函数可以由每个段的完全不同类型的函数组成。在新的样条回归的基础上，提供了一种估计分布和密度函数的新方法，该方法在数值实验中显示出优于现有方法的显着优势。最后，讨论了高维随机变量的密度函数估计。