| 摘要: |
| 目的 针对一元随机变量的概率密度函数估计问题,寻找一个提高非参数估计收敛速度的方法。 方法 通过
对参数估计和非参数估计组合加权,提出一个基于正则惩罚的半参数估计方法:假定一个参数模型得到其对应的
参数估计,用核方法计算得到其非参数估计,通过带正则惩罚的经验似然损失或积分平方损失得到一个权重,基于
这个权重对得到的参数估计和非参数估计组合,从而得到所提出的基于正则惩罚的半参数估计。 结果 渐近理论性
质显示:所提出的带正则惩罚的半参数密度估计方法结合了参数估计和非参数估计的优点,不依赖任何模型假设,
在任何情况下都是收敛的,并且在参数模型假设正确时,权重偏向参数估计,此时收敛速度与参数估计一样,反之
则偏向非参数估计,收敛速度与非参数估计一致。 通过数值模拟实验发现:当数据满足参数模型假设时,带惩罚的
半参数密度估计方法权重偏向参数估计,与理论结果一致;当数据不满足参数模型假设时,所提出的半参数估计权
重偏向非参数估计,这同样与理论结果一致。 最后将该方法应用于重庆市降水数据中,研究了其月降水量的分布。
结论 实例分析结果表明:所提出的基于正则惩罚的半参数估计与非参数估计相比更光滑,与参数估计相比拟合更
好,验证了该方法的合理性。 |
| 关键词: 密度函数估计 参数估计 非参数估计 加权 正则惩罚 |
| DOI： |
| 分类号: |
| 基金项目: |
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| Semi-parametric Density Estimation Method Based on Regular Penalty |
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TAN Xin1 YAN Mei2
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1. School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu 511731 China
2. School of Mathematics Yunnan Normal University Kunming 650500 China
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| Abstract: |
| Objective This paper aimed to find a method to improve the convergence speed of nonparametric estimation for
the probability density function of a univariate random variable. Methods By combining parameter estimation and
nonparametric estimation with weighting a semiparametric estimation method based on regular penalty was proposed. In
this method a parameter model was assumed to obtain its corresponding parameter estimation. Nonparametric estimation
was computed using kernel methods. A weight was then obtained through empirical likelihood loss with a regular penalty or
integral square loss. Based on this weight the proposed semiparametric estimation based on regular penalty was obtained
by combining the parameter estimation with nonparametric estimation. Results Asymptotic theoretical properties show that
the proposed semiparametric estimation method based on regular penalty combines the advantages of parameter estimation
and nonparametric estimation. It converges in any case without relying on any model assumption and when the parameter
model assumption is correct the weight tends toward the parameter estimation resulting in the same convergence speed as the parameter estimation otherwise it tends toward the nonparametric estimation with consistent convergence speed as
nonparametric estimation. Numerical simulation experiments show that when the data satisfy the parameter model
assumption the weight of the penalized semi-parametric density estimation method tends toward the parameter estimation
consistent with the theoretical results when the data do not satisfy the parameter model assumption the weight of the
proposed semi-parametric estimation tends toward the nonparametric estimation also consistent with the theoretical
results. Finally this method was applied to precipitation data in Chongqing to study the distribution of monthly
precipitation. Conclusion The results of the case study show that the proposed semiparametric density estimation method
based on the regular penalty is smoother compared with nonparametric estimation and fits better compared with parameter
estimation thus validating the rationality of this method. |
| Key words: density function estimation parameter estimation nonparametric estimation weighting regular penalty |
