Almost Stochastic Dominance and Moments

This paper first extends the theory of almost stochastic dominance (ASD) to the first four orders. We then establish some equivalent relationships for the first four orders of the ASD. Using these results, we prove formally that the ASD definition modified by Tzeng et al.\ (2012) does not possess any hierarchy property. Thereafter, we conclude that when the first four orders of ASD are used in the prospects comparison, risk-averse investors prefer the one with positive gain, smaller variance, positive skewness, and smaller kurtosis. This information, in turn, enables decision makers to determine the ASD relationship among prospects when they know the moments of the prospects. At last, we discuss the necessary and sufficient conditions for different orders the ASD and the moments of the prospects.