Tail-restricted stochastic dominance

We introduce here a class for stochastic dominance between two probability distributions for which we have chosen the name tail-restricted stochastic dominance. As its name suggests, it is concerned with dominance restricted to a range of values of the random variables, which disregards very unlikely outcomes, as those in the tail(s) of the distributions. The probability for the occurrence of these unlikely values, which is defined as the tail restriction probability value, is proposed as a measure for the restriction. We discuss the differences it guards with two other restricted dominance concepts: the Leshno and Levy's almost stochastic dominance for investment decisions (restricted on the shape of the utility function) and the Davidson and Duclos' restricted stochastic dominance in poverty studies (restricted on a threshold for poverty level). We illustrate its application in an investment decision under risk with real data and provide a simple and straightforward procedure based on a quantile approach for verifying the existence of dominance (either restricted or unrestricted) in the general case of two empirical distributions. We also apply the concept in the analysis of the St. Petersburg paradox.