# Questions tagged [stochastic-ordering]

A stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders.

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### Probability that a sample drawn from one distribution is lower than a sample drawn from another distribution?

Context: we don't know the exact distribution parameters, however in practice we can obtain many samples from each distribution. Case 1: let's say that I have a sample of size N from each distribution....
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### How to put priors on permutation matrices for "permutation regression"?

Background I would like to put prior probabilities over permutation matrices $\mathbf{P}$ $$\vec A = \mathbf{P} \vec B$$ where $\vec A$ and $\vec B$ are the same random vectors up to a permutation (...
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### Suppose we know $\mu_X<\mu_Y$. How can that information be used to improve estimates of the mean of $X$ and $Y$

Suppose there are two distributions which we know nothing about except that the mean of second is greater than the first and each can be sampled. Intuitively, the estimate of both means should be able ...
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### First order stochastic ordering implies countably additive probability measure?

Let $P$ be a finitely additive probability measure. I learn from my friend that: $[P(X>Y)=1 \implies \mathbb E_P[X]>\mathbb E_P[Y]]\iff$ $P$ is countably additive. Seems to be a very useful ...
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### Equivalent definition of stochastic dominance

Note that a distribution function (cadlag etc) $F$ is said to be stochastically dominated by a distribution function $G$ if $F(x)\geq G(x)$ for all $x \in \mathbb{R}$. The following result ...
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### What is a rank?

I've been thinking about unifying some notions related to ranking, order theory, ordinal data, and graded posets. While the notion of a grade in order theory is quite general, in some sense the way we ...
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### Are these equivalent (for p-values): super-uniform, stochastically larger than / dominating the uniform, conservative?

In the literature and online, I have found three different wordings that I think refer to the same concept: stochastically larger than uniform (which I take is ...
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### Two distributions, same mean, different variance: Stochastic dominance for deviation from mean?

Say you have two (bounded) random variables, $X$ and $Y$, on the same discrete probability space, such that $E(X)=E(Y)$ but $Var(X) < Var(Y)$. Do I know that, for any $k \geq 0$,  \text{Prob}(|X-...
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### Are unbiased efficient estimators stochastically dominant over other (median) unbiased estimators?

General description Does an efficient estimator (which has sample variance equal to the Cramér–Rao bound) maximize the probability for being close to the true parameter $\theta$? Say we compare the ...
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### Stochastic order does not imply (nor is implied by) mean residual life order

Let $X,Y$ be nonnegative random variables with cummulative distribution functions $F$ and $G$ respectively. Then $X\le_{\text{st}}Y$ iff $1-F(x)\le 1-G(x)$ for all $x\ge 0$ annd $X\le_{\text{mrl}}Y$ ...
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### Why care so much about expected utility?

I have a naive question about decision theory. We calculate the probabilities of various outcomes assuming particular decisions and assign utilities or costs to each outcome. We find the optimal ...
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### Second Order Stochastic Dominance and $\Pr[X \leq Y]\leq1/2$

Let $X$ be a random variables with continuou cdf $F$ in the support $[a,b]$. Let $Y$ be another r.v. with cdf $G$, which is continuous in $[a,b)$ and has an atom at $b$. $X$ and $Y$ are independent. ...
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### Stochastic ordering

I'm a bit stuck with the following. Assume $X$ ~$N(0,1)$ and $Y$~$N(1,1)$. Is $Y$ stochastic bigger than $X$? And the same question goes for $X$~$N(0,1)$, $Y$~$N(0,4)$.. I simply don't have a clue ...
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### X is stochastically increasing in Y $\implies$ $E\left[Y| X\right]$ increasing in X

I have two random variables, $X$ and $Y$. I know: $\text{pr}\left(X \le u | Y\right)$ is a decreasing function of $Y$ for all $u$. Does this imply that: $\mathbb{E}\left[Y | X\right]$ is increasing ...
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### How can I prove stochastic dominance if I don't know the CDF?

Is there a smart way to show that one variable stochastically dominates the other without knowledge of the CDF? Thanks so much!
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### Characteristic functions can establish stochastic dominance?

In an answer to the question here What is the purpose of characteristic functions? people answered the general question about characteristic functions. One answer mentioned that one can use it to ...
Suppose we have two distributions $F$ and $G$ over $\left[0,1\right]$. Suppose $F(x) \leq G(x)$ for all $x$, i.e. $F$ first-order stochastically dominates $G$. Is it true that \$F(x|x\leq k) \leq G(x|x\...