Bias in Nonlinear Estimation
Bias in Nonlinear Estimation
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DOI：
10.1111/j.2517-6161.1971.tb00871.x
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发表时间：
1971-07
期刊：
影响因子：
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通讯作者：
M. J. Box
中科院分区：
文献类型：
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作者：
M. J. Box
虽然人们普遍认为，最大似然估计的非线性模型中的参数通常是有偏见的，似乎很少有工作已经做了定量评估这些偏见。本文首先通过一个简单的例子说明了精确计算偏差的困难，然后给出了一类非线性最小二乘问题偏差的一般计算方法。贝叶斯估计中的偏差也被考虑，尽管说明性的例子都是针对均匀先验的情况，即估计是最大似然的。在最重要的辅助结果中，定义了参数估计的偏差与方差-协方差矩阵的广义比，并表明与Beale（1960）的非线性度量密切相关。最后的哲学被应用到一个有点不同的问题，即确定的伽玛分布的参数的最大似然估计的偏差。
Although it is widely recognized that maximum-likelihood estimates of the parameters in non-linear models are generally biased, little work appears to have been done on quantitatively assessing these biases. In this paper the difficulties of exact calculation of the bias for a simple example are first illustrated, after which a general method of calculating the biases in a class of nonlinear least-squares problems is presented. The bias in Bayesian estimation is also considered, although the illustrative examples are all for the case of a uniform prior, i.e. the estimation is maximum likelihood. In the most important of the subsidiary results, a generalizedratio of the bias to the variance-covariance matrix of the parameter estimates is defined, and shown to be closely related to Beale's (1960) measures of nonlinearity. Finally the philosophy is applied to a somewhat different problem, namely determining the biases in the maximum-likelihood estimates of the parameters of the gamma distribution.