Comparison of Probability Distributions
Comparison of Probability Distributions
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概率分布比较
DOI：
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发表时间：
1974
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通讯作者：
J. Lindsey
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文献类型：
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作者：
J. Lindsey
通常，在考虑实验的统计模型时，不止一个概率分布在理论上是可行的。对于所有观测都是在相同的响应条件下进行的简单情况，我们将讨论用似然方法（例如，参见Sprott和Kalbfleisch，1969）确定更合理的分布的问题。（Lindsey，1974，将在独立变量存在时考虑这个问题。为了使用似然推断来做到这一点，必须引入一个基础统计模型，可以与所考虑的所有其他分布进行比较。下面的推导得出多项式模型作为基础模型。在文献中已经提出了几种方法来确定一些可能的模型中哪一个最好地描述了一组数据。考克斯（1961，1962）发展了渐近Neyman-Pearson似然比检验，并提出了另一种方法，包括密度函数的加法或乘法组合，以及附加参数的估计。Atkinson（1970）进一步发展了这一方法。当每个模型和模型内参数的先验概率都可用时，Lindley（1961，第456页）使用贝叶斯定理给出了两个模型的后验比值比。在适用的情况下（即当先验概率可用时），这种方法可以与下面开发的方法一起使用。
OFTEN, more than one probability distribution is theoretically feasible when considering statistical models for an experiment. The problem of determination of the more plausible distribution using likelihood procedures (see, for example, Sprott and Kalbfleisch, 1969) will be discussed for the simple case where all observations are made under the same response conditions. (Lindsey, 1974, will consider this problem when independent variables are present.) To do this using likelihood inference, a base statistical model must be introduced with which all other distributions under consideration may be compared. The derivation which follows yields the multinomial model as the base model. Several approaches have been suggested in the literature to the problem of determining which of a number of possible models best describes a set of data. Cox (1961, 1962) develops asymptotic Neyman-Pearson likelihood ratio tests and suggests an alternative approach involving a combination, either additive or multiplicative, of the density functions, with estimation of additional parameters. This approach is further developed by Atkinson (1970). When prior probabilities, both for each model and for the parameters within the models, are available, Lindley (1961, p. 456) gives a posterior odds ratio of the two models using Bayes's theorem. When applicable (i.e. when prior probabilities are available), this approach may be used with the methods developed below.