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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Learn probablistic rank order from lots paired comparisons
Suppose I have $k$ different unknown probability distributions (e.g. $A$, $B$, ...) and I have data that provide examples of draws from pairs of distributions (e.g. $a_1$ and $c_1$, then $b_2$ and $...
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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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Proof of likelihood ratio ordering implying hazard rate ordering
I'm studying stochastic orders and am stuck in the proof of likelihood ratio ordering implying hazard rate ordering.
Definitions: Let $X$ and $Y$ be two random variables with respective densities $f(...
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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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Class of density functions that p(ax)/p(x) is non-decreasing for a<1
What are the probability density functions $p_X(x)$, with support $\subset [0,\infty)$, that for all $a<1$, $\frac{p_X(ax)}{p_X(x)}$ is a non-decreasing function of $x$ over the support of $P_X (x)...
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Class of density functions that f(x-c)/f(x) is non-increasing
I am looking for a class of probability density functions $f_X(x)$, with support $\subset [0,\infty)$, for which there exist a non-empty subset of $\mathbb{R}^+$ denoted by $S_X$, such that $\forall ...
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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 ...
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Probability of uniformly drawing N numbers less than the expected second highest value
In the case of 3 draws (N=3) from Uniform[0,1], the expected second highest value would be 1/2. Although unlikely it could happen that all three numbers were less than 1/2. It is exactly this ...
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First-order stochastic dominance and truncations
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\...
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