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able to rank concept of stochastic confidence level consistency of risk consistent with expected consistent with second Core Inflation definition of tail degree distribution denoting the profit dominance is defined dominates X2 elimination of tail elliptical distribution expected shortfall expected utility maximization extreme loss following holds free of tail Hiroshi Fujiki Inflation Ingersoll 1987 Kunio Okina larger for portfolio level of threshold Levy and Kroll Market Liquidity maximization and elimination maximization and free Monetary Policy N-th order Naohiko Baba Number order distribution function order stochastic dominance order tail risk Pareto distribution partial information payoff portfolios are ranked profit and loss random variables denoting ranked by second risk measure consistent risk measure p(X second lower partial second order stochastic second order tail Shigenori Shiratsuka shortfall is consistent shows that expected Student's t-distribution sub-additive th lower partial th order stochastic Toshinao Yoshiba underlying distributions utility functions VaRa(X Xx and X2 Yasuhiro Yamai