# Sets Closed under Stochastic Dominance Ordering

I'm working on a problem involving stochastic dominance and minimums'' of sets of random variables.

For concreteness, consider two distributions with cdfs $F(x)$ and $G(x)$. We say that $F$ first-order stochastically dominates $G$ if $F(x) \leq G(x)$ for all $x$.

Consider the set $S$ of non-negative random variables that all have the same median $m$. Even though this set has two properties that I'm interested in

1. it is very natural'', in the sense that it contains all distributions coherent with some partial information.

2. it has a minimum'' under stochastic dominance: the discrete random variable which assigns probability $50 \%$ to 0 and probability $50 \%$ to $m$. This random variable belongs in $S$, and is dominated by all other random variables in $S$.

My question is, are there other natural'' sets that have a minimum under stochastic dominance? For example, if I take the set $T(\mu,\sigma)$ of all positive random variables with mean $\mu$ and variance $\sigma^2$, does this set have a minimum under stochastic dominance?

Thanks.

• It seems clear that $F(\mu)$ can be arbitrarily close to $1$, but cannot be $=1$ (unless we're in the trivial case $\sigma=0$), so there are no minimal distributions in this sense. – Christian Remling Aug 10 '14 at 21:22
• Please have a look at my question, it seems they are somehow related. – Seyhmus Güngören May 10 '16 at 0:02

(1)Consider the set U of non-negative random variables such that for all random variables, we have $$sup\{x: P(X\le x)\le u\}=m, \,\,\,\,\,\, u \in(0,1)$$ "the discrete random variable which assigns probability u% to 0 and probability (1-u)% to m." \ (2): Let the random variable $X_{(\lambda_1,\cdots,\lambda_n)}$ such that: $$p(X_{(\lambda_1,\cdots,\lambda_n)}\le x)=\prod_{i=1}^n (1-e^{\lambda_ix}),\,\,\,\,\,x\ge 0 \& \lambda_i \ge 0$$ Consider the set $U=\{X_{(\lambda_1,\cdots,\lambda_n)}:\sum_{i=1}^n \lambda_i=m,\,\,\,or \,\,\prod_{i=1}^n \lambda_i=p\}$ . \ "The random variable $X_{(p,...,p)}\in U$ and $P (X_{(p,...,p)}\le x)\ge P(X_{(\lambda_1,\cdots,\lambda_n)}\le x)$ for all $X_{(\lambda_1,\cdots,\lambda_n)}\in U$."