On the (Im-)Possibility of Representing Probability Distributions as a Difference of I.I.D. Noise Terms
研究了随机变量能否表示为两个独立同分布随机项之差的问题，发现此类变量的密度函数不能近似均匀、拟凸或严格凹，且即使分量光滑也常出现扭结，警示经济学中直接对噪声项之差施加假设的策略可能导致模型不一致。
A random variable is difference-form decomposable (DFD) if it may be written as the difference of two i.i.d. random terms. We show that densities of such variables exhibit a remarkable degree of structure. Specifically, a DFD density can be neither approximately uniform, nor quasiconvex, nor strictly concave. On the other hand, a DFD density need, in general, be neither unimodal nor logconcave. Regarding smoothness, we show that a compactly supported DFD density cannot be analytic and will often exhibit a kink even if its components are smooth. The analysis highlights the risks for model consistency resulting from the strategy widely adopted in the economics literature of imposing assumptions directly on a difference of noise terms rather than on its components.