用于Tweedie位置、尺度和形状模型的平滑参数估计的广义Fellner-Schall方法
A generalized Fellner-Schall method for smoothing parameter estimation with application to Tweedie location, scale and shape models
摘要：
本文考虑了具有二次惩罚系数的正则对数似然模型中平滑参数和方差分量的估计，推广了Fellner（1986）和Schall（1991）的方法。特别是：（i）我们将原始方法推广到惩罚在几个平滑参数中是线性的情况，从而覆盖张量积和自适应平滑器的重要情况;（ii）我们展示了为什么该方法的步骤增加了模型的受限边际似然，它往往比期望最大化算法收敛得更快，或者明显加速，并研究其与牛顿优化的关系;（iii）我们将该方法推广到任何Fisher正则似然。该方法代表了一个相当大的简化，在现有的方法估计平滑参数的背景下，定期似然，而不牺牲一般性：例如，它只需要计算相同的第一和第二导数的对数似然系数估计所需的，而不是与第三或第四阶导数所需的替代方法。提供的例子，这将是不可能的或不切实际的预先存在的Fellner-Schall方法，以及一个例子的Tweedie的位置，规模和形状模型，这将是一个挑战的替代方法。
英文摘要：
We consider the estimation of smoothing parameters and variance components in models with a regular log likelihood subject to quadratic penalization of the model coefficients, via a generalization of the method of Fellner (1986) and Schall (1991). In particular: (i) we generalize the original method to the case of penalties that are linear in several smoothing parameters, thereby covering the important cases of tensor product and adaptive smoothers; (ii) we show why the method's steps increase the restricted marginal likelihood of the model, that it tends to converge faster than the EM algorithm, or obvious accelerations of this, and investigate its relation to Newton optimization; (iii) we generalize the method to any Fisher regular likelihood. The method represents a considerable simplification over existing methods of estimating smoothing parameters in the context of regular likelihoods, without sacrificing generality: for example, it is only necessary to compute with the same first and second derivatives of the log-likelihood required for coefficient estimation, and not with the third or fourth order derivatives required by alternative approaches. Examples are provided which would have been impossible or impractical with pre-existing Fellner-Schall methods, along with an example of a Tweedie location, scale and shape model which would be a challenge for alternative methods.
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