# 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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### Can two different measures have the same first order stochastic dominance?

$(S,\Sigma,\mu)$ is the common probability triple, where $S=[0,1]$, $\Sigma$ is the Borel sigma algebra, and $\mu$ is the Lebesgue measure. $X:[0,1]\to\mathbb R$ is a r.v. There are two different ...
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### Is Kruskal-Wallis really a good test of stochastic superiority?

After much reading/googling, I reached the conclusion that the Kruskal-Wallis test (K-W) is a test of stochastic superiority (or, equivalently, of stochastic equivalence). Rejecting the null implies ...
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### Lower Bound on Expected Maximum and Upper Bound on Expected Minimum of Order Statistics

This question relates to bounds on expectations of order statistics, elaborated upon in the Book by Arnold and Balakrishnan (1989). Let $X_1,\ldots,X_n$ be i.i.d. continuous random variables ...
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### Can we rank markov matrices in stochastic order?

I am familiar with the concept of stochastic ordering for two random variables and how we can say if a markov matrix is stochastically monotone. What im interested in is if there is a concept for ...
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### "Second order correct", "first order asymptotics"

I keep seeing phrases like "first order asymptotic", "second order correct", "high order asymptotics" and I am honestly don't know how these terms are rigorously/...
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### Are sample means ordered by strict second-order stochastic dominance throughout the support?

Consider random variables $X_1,X_2,\dots$. Each $X_i$ is independent and identically distributed on $[0,1]$ with a cumulative distribution $F$ that has a positive density $f(x)>0$ throughout the ...
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### Order Statistics with joint density

I have an guess in a larger stochastic problem. I assume following: Let $x,y$ be two variables, with $y<x$ and let $f(\cdot)$, $F(\cdot)$ be the a continous PDF and CDF with support $[0,z]$. I ...
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### Does a general applicable transitive relation exist between test statistics and p-values?

For a proof I showed that, if a p-value has a uniform distribution between 0 and 1, then transforming these p-values back using the inverse of standard normal results in a random variable with ...
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### Total (stochastic) orders for univariate distributions on compact set

Let $[\underline{x},\bar{x}] = X \in\mathbb{R}_{+}$ be a compact set and $\mathcal{F}$ be the set of all CDFs $F$ on $X$. What total stochastic orders can I define on $\mathcal{F}$? (see Wikipedia ...
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### Stochastic Dominance for convex sum of two random variables with same distribution

This is a very widely used result in finance and economics, and seems fairly intuitive as well (riskier asset is less preferred by risk-averse (concave utility) investors). However, I have not been ...
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### What is it called when a random variable is weakly greater than another for all elements of the sample space?

Suppose I have random variables $(X_1,X_2)$ defined on a probability space $(\Omega, \mathcal{F},P)$ such that for any element $\omega \in \Omega$, $X_1(\omega) \geq X_2(\omega)$. I'm looking to work ...
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### Stochastic Comparability

I have ran into a paper where the author speaks of finding the variance for the AUC (Area Under the ROC curve) for two continuous variables X and Y, which are stochastically comparable . Does anybody ...
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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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### 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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### R: How can I represent partially-ordered time series in R?

I believe that this is a statistics rather than a programming question, though I am tied to an R implementation and hope for a reply in kind. I have data that constitutes several time series. I ...
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### Testing against non-dominance of discrete distributions

I have two discrete distributions $A$ and $B$ with independent draws. What tests can I use against $H_0$: $A$ does not first-order stochastically dominate $B$?
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