The "Devil's Horns" Problem of Inverting Confluent Characteristic Functions
警告反演合流特征函数时可能出现的一类问题，用二元情形和简单自回归模型例子说明联合退化如何扭曲边际密度的推导，对从事特征函数反演和分布推导的研究者有参考价值。
WE WARN OF A CLASS of problems that can occur when inverting confluent characteristic functions (CF's). The term confluence is often used in Mathematics in connection with analysis and/or dynamic (difference, differential, and integral) equations; for example, see the classic text by Whittaker and Watson (1927). A confluence (of singularities) is a joint degeneracy that occurs within a function; here, the CF. When one is dealing with the CF of a k-dimensional variate where k > 1, these joint degeneracies can distort the derivation of the marginal density of some lower-dimensional combination of the k components. The distortions are both analytical and numerical. In this note, we first express the distributional problem in the simplest bivariate case, then clarify it with examples from a simple autoregressive (AR) model. Let R, S be two continuous (for simplicity) variates based on a sample of n observations, with joint CF pn(u,v) =E[euR+ivS], i = , and Pr{S > 0} = 1. The joint density h,jr, s) of R and S is expressed by means of the inversion formula as