Sets Closed under Stochastic Dominance Ordering

Question

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.

Answer 1

(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: \begin{equation}p(X_{(\lambda_1,\cdots,\lambda_n)}\le x)=\prod_{i=1}^n (1-e^{\lambda_ix}),\,\,\,\,\,x\ge 0 \& \lambda_i \ge 0\end{equation} 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 $X_{(\lambda_1,\cdots,\lambda_n)}\in U$."