# Questions tagged [stochastic-ordering]

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### When to test For Equality of Medians, and when Stochastic Equality?

In statistics, we are very often interested in investigating whether some score (let's say life satisfaction) tends to be bigger in one population (let's say rich people) compared to another ...
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### Does the mean preserving spread of a distribution constitute a mean preserving spread of the joint distribution of two iid draws from it?

Let random vectors $X_1, X_2 \sim F, \;i.i.d, X_1, X_2 \in X$. Now replace $F$ with its mean-preserving spread (MPS), say $G$. My question is, does that constitute an MPS of the joint distribution of ...
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### Stochastic dominance and mean preserving spread

I need someones help on understanding the concepts of stochastic dominance and mean preserving spread. I have an exercise which could be used for explanation. Consider the following lotteries: L1 ={...
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### First Order Stochastic Dominance

I'm trying to solve an exercise about first order stochastic dominance, and there is this one question that I'm not able to answer. Here are the hypothesis of the exercise: Let Y ∼ F and Y' ∼ G, ...
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### Kruskal-Wallis Test: Identically Shaped Distributions

I'm currently working with the Kruskall-Wallis test and had a question about what people mean when they speak about 'Identically Shaped Distributions' and how I could identify whether my data fits ...
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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\...