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点估计

统计学中,点估計(英語:point estimation)是指以样本数据来估计总体母數, 估计结果使用一个点的数值表示“最佳估计值”,因此称为点估计。由样本数据估计总体分布所含未知参数的真實值,所得到的值,称为估计值。

点估计可以与区间估计形成对比:这种区间估计通常是在频率论推断的情况下的置信区间 ,或在贝叶斯推断的情况下的可信区间 。

估计法

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目前有多种估计法可供选择,每种估计法都有不同属性。

贝叶斯点估计

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贝叶斯推断通常基于后验分布 。 许多贝叶斯估计量是后验分布的集中趋势统计量,例如,它的均值,中位数或模式:

  • 后均值 ,最小化平方误差损失函数的(后验) 风险 (预期损失);在贝叶斯估计中,风险是根据高斯观察到的后验分布来定义的。 [1]
  • 后验中位数 ,最小化绝对值损失函数的后验风险,如拉普拉斯所观察到的。 [1] [2]
  • 最大后验MAP ),其发现最大的后验分布;对于统一的先验概率,MAP估计量与最大似然估计一致;

MAP估计具有良好的渐近性质,对于许多复杂问题,最大似然估计也存在局限性。 对于最大似然估计符合一致性的常规问题,最大似然估计的最终结果与MAP估计一致。 [3] [4] [5] 根据瓦尔德定理,贝叶斯估计是可以接受的。 [4] [6]

最小消息长度 ( MML )点估计基于贝叶斯信息理论 ,并不与后验分布直接相关。

贝叶斯滤波器存在以下特殊情况:

以下几种计算统计迭代法与贝叶斯分析有密切联系:

点估计的属性

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參見

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参考文献

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  1. ^ 1.0 1.1 Dodge, Yadolah (编). Statistical data analysis based on the L1-norm and related methods: Papers from the First International Conference held at Neuchâtel, August 31–September 4, 1987. Amsterdam: North-Holland Publishing Co. 1987. 
  2. ^ Jaynes, E.T. Probability theory : the logic of science 5. print. Cambridge [u.a.]: Cambridge Univ. Press. 2007: 172. ISBN 978-0-521-59271-0. 
  3. ^ Ferguson, Thomas S. A course in large sample theory. Chapman & Hall. 1996. ISBN 0-412-04371-8. 
  4. ^ 4.0 4.1 Le Cam, Lucien. Asymptotic methods in statistical decision theory. Springer-Verlag. 1986. ISBN 0-387-96307-3. 
  5. ^ Ferguson, Thomas S. An inconsistent maximum likelihood estimate. Journal of the American Statistical Association. 1982, 77 (380): 831–834. JSTOR 2287314. doi:10.1080/01621459.1982.10477894. 
  6. ^ Lehmann, E.L.; Casella, G. Theory of Point Estimation, 2nd ed. Springer. 1998. ISBN 0-387-98502-6. 

扩展阅读

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  • Bickel, Peter J. & Doksum, Kjell A. Mathematical Statistics: Basic and Selected Topics I Second (updated printing 2007). Pearson Prentice-Hall. 2001. 
  • Lehmann, Erich. Theory of Point Estimation. 1983. 
  • Liese, Friedrich & Miescke, Klaus-J. Statistical Decision Theory: Estimation, Testing, and Selection. Springer. 2008. 
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点估计
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