This paper presents some new stochastic dominance (SD) criteria for ranking transformations on a random variable, which is the first time that this is done for transformations under the discrete framework. By using the expected utility theory, the authors first propose a sufficient condition for general transformations by first degree SD (FSD), and further develop it into the necessary and sufficient condition for monotonic transformations. For the second degree SD (SSD) case, they divide the monotonic transformations into increasing and decreasing categories, and further derive their necessary and sufficient conditions, respectively. For two different discrete random variables with the same possible states, the authors obtain the sufficient and necessary conditions for FSD and SSD, respectively. The new SD criteria have the following features: each FSD condition is represented by the transformation functions and each SSD condition is characterized by the transformation functions and the probability distributions of the random variable. This is different from the classical SD approach where FSD and SSD conditions are described by cumulative distribution functions. Finally, a numerical example is provided to show the effectiveness of the new SD criteria.
JEL C51 D81Keywords Stochastic dominance; transformation; utility theory; insurance
AuthorsJianwei Gao, School of Economics and Management, North China Electric Power University, Beijing, China, gaojianwei111@sina.com Feng Zhao, School of Economics and Management, North China Electric Power University, Beijing, China Citation Jianwei Gao and Feng Zhao (2017). A New Approach of Stochastic Dominance for Ranking Transformations on the Discrete Random Variable. Economics: The Open-Access, Open-Assessment E-Journal, 11 (2017-14): 1-23. http:// dx.doi.org/10.5018/economics-ejournal.ja. Economics: The Open-Access, Open-Assessment E-Journal 11 (2017-14) www.economics-ejournal.org 2
IntroductionIn real world, many human activities in insurance and financial fields induce risk transformations. For example, we assume that an investor owns a house where X denotes the value of the house (a random variable). The investor can insure the house with various levels of deductions. By choosing two different deduction policies, the investor creates different transformations ( ) m X and ( ) n X . Then an interesting question occurs: which deduction policy (transformation) dominates the other? In other words, how to find an effective approach for ranking these transformations so as to choose the beneficial one? Stochastic dominance (SD) is one of the most famous approaches to comparing pairs of prospects. Presented in the context of expected utility theory, the SD approach has the advantage that it requires no restrictions on probability distributions. Well-known specifications of SD are first degree SD (FSD) and second degree SD (SSD), which by far attract most of the attention in SD research. Due to the advantage mentioned above, the SD approach has been proved to be a po...