The Use of Maximum Likelihood Estimates in {\chi^2} Tests for Goodness of Fit
The Use of Maximum Likelihood Estimates in {\chi^2} Tests for Goodness of Fit
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DOI：
10.1007/978-1-4614-1412-4_47
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发表时间：
1954-09
影响因子：
--
通讯作者：
H. Chernoff;E. Lehmann
中科院分区：
文献类型：
--
作者：
H. Chernoff;E. Lehmann
通常的检验是样本来自给定形式的分布，方法是计算落入指定单元的观测值的数量，并对这些频率进行χ2检验。在估计该测试的参数时，可以使用基于(1)小区频率或(2)原始观测的最大似然(或等效)估计。本文证明了，与文献(1)的结果不同，文(2)中的检验统计量不具有极限χ~2分布，而是随机地大于χ~2理论下的期望分布。给出了极限分布，并进行了算例计算。这表明，对于泊松分布的拟合，误差不是很严重，但对于正态分布的拟合，误差可能是很大的。
The usual test that a sample comes from a distribution of given form is performed by counting the number of observations falling into specified cells and applying the χ2test to these frequencies. In estimating the parameters for this test, one may use the maximum likelihood (or equivalent) estimate based (1) on the cell frequencies, or (2) on the original observations. This paper shows that in (2), unlike the well known result for (1), the test statistic does not have a limiting χ2-distribution, but that it is stochastically larger than would be expected under the χ2theory. The limiting distribution is obtained and some examples are computed. These indicate that the error is not serious in the case of fitting a Poisson distribution, but may be so for the fitting of a normal.