Bayesian estimation for nonparametric regression
Received date: 2017-02-15
Online published: 2018-12-26
基于因变量 $Y$ 对自变量 ${X}$ 条件分布的非参数贝叶斯估计, 通过期望计算得到未知回归函数的后验估计表达式, 并计算出估计的均方误差, 证明该估计的均方收敛性. 阐明当先验的选择接近真实的回归函数时, 该估计的均方误差小于局部线性核回归的均方误差. 最后通过实证分析, 表明该非参数贝叶斯回归比非参数局部线性回归具有更好的预测效果.
关键词： 非参数贝叶斯回归; 非参数贝叶斯分布估计; Dirichlet过程; 局部线性回归; 人口预测
苏雅玲, 何幼桦 . 非参数回归的贝叶斯估计[J]. 上海大学学报(自然科学版), 2018 , 24(6) : 1022 -1029 . DOI: 10.12066/j.issn.1007-2861.1926
Based on nonparametric Bayesian estimation of a conditional distribution, posterior estimation of an unknown regression function is obtained by calculating its expectation. The mean square error of the estimation is calculated. Its convergence in mean square of the estimation is proved. It is shown that the mean square error of the estimation is less than that of the local linear kernel regression when prior regression is chosen to be close to the unknown regression function. Empirical evidence shows that the nonparametric Bayesian regression may be more effective in prediction than local linear regression.
| [1] | Devroye L. The uniform convergence of the Nadaraya-Watson regression function estimate[J]. Canadian Journal of Statistics, 1978,6(2):179-191. |
| [2] | Greblicki W, Pawlak M. Distribution-free pointwise consistency of kernel regressionestimate[J]. J Annals of Statistics, 1984,12(4):1570-1575. |
| [3] | Devroye L. On the almost everywhere convergence of nonparametric regression function estimates[J]. J Annals of Statistics, 1981,9(6):1310-1319. |
| [4] | Fan J Q. Local linear regression smoothers and their minimax efficiencies[J]. J Annals of Statistics, 1993,21(1):196-216. |
| [5] | Antoniadis A, Gregoire G, McKeague I W. Wavelet methods for curve estimation[J]. Journal of the American Statistical Association, 1994,89(428):1340-1353. |
| [6] | Ferguson T. A Bayesian analysis of some nonparametric problems[J]. J Annals of Statistics, 1973,1(2):209-230. |
| [7] | Jayaram S. A constructive definition of Dirichlet priors[J]. J Statistics Sinica, 1994,4(2):639-650. |
| [8] | 姚宗静. 基于 Dirichlet 过程的非参数贝叶斯分析[D]. 成都: 西南交通大学, 2007. |
| [9] | 龙杏芬, 张德生, 武新乾. 非参数回归模型的贝叶斯局部线性估计[J]. 山西大学学报(自然科学版), 2010,33(3):371-374. |
| [10] | 卢一强, 茆诗松. 广义非参数回归的 B 样本贝叶斯估计[J]. 应用数学, 2015,18(1):8-13. |
| [11] | 沃塞曼. 现代数学译丛 3: 现代非参数统计 [M]. 北京: 科学出版社, 2008: 58-65. |
| [12] | Hardle W, Werwatz A, Müller M, et al. Nonparametric and semiparametric models[J]. Springer Series in Statistics, 2006,62(2):628-629. |
| [13] | Ruppert D, Wand M P. Multivariate locally weighted least squares regression[J]. Annals of Statistics, 1994,22(3):1346-1370. |
| [14] | 王学保, 蔡果兰. Logistic 模型的参数估计及人口预测[J]. 北京工商大学学报(自然科学版), 2009,27(6):75-78. |
/
| 〈 |
|
〉 |
