Series evaluation of Tweedie exponential dispersion model densities
Series evaluation of Tweedie exponential dispersion model densities
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DOI：
10.1007/s11222-005-4070-y
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发表时间：
2005-10-01
影响因子：
2.2
通讯作者：
Gordon, KS
中科院分区：
文献类型：
--
作者：
Dunn, PK;Gordon, KS
指数离差模型是广义线性模型的原型响应分布，它是具有离差参数的线性指数族。Tweedie族包括具有幂均值-方差关系的指数离差模型。正态分布、泊松分布、伽马分布和逆高斯分布属于Tweedie族。除了这些特殊情况，特威迪分布没有密度函数，可以写在封闭形式。相反，密度可以表示为从级数展开导出的无限求和。本文描述了如何以数值有效的方式对级数展开求和。证明了该方法的实用性，但完整的机器精度不被证明是可获得的所有参数值使用级数展开法。相对于分散参数的密度的导数也推导出，以便于最大似然估计。两个数据的例子，并与Box-Cox变换和扩展的准似然方法进行了比较。
Exponential dispersion models, which are linear exponential families with a dispersion parameter, are the prototype response distributions for generalized linear models. The Tweedie family comprises those exponential dispersion models with power mean-variance relationships. The normal, Poisson, gamma and inverse Gaussian distributions belong to the Tweedie family. Apart from these special cases, Tweedie distributions do not have density functions which can be written in closed form. Instead, the densities can be represented as infinite summations derived from series expansions. This article describes how the series expansions can be summed in an numerically efficient fashion. The usefulness of the approach is demonstrated, but full machine accuracy is shown not to be obtainable using the series expansion method for all parameter values. Derivatives of the density with respect to the dispersion parameter are also derived to facilitate maximum likelihood estimation. The methods are demonstrated on two data examples and compared with with Box-Cox transformations and extended quasi-likelihoood.