Questions tagged [stochastic-ordering]

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1answer
103 views

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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0answers
10 views

Is the Beta Distribution ranked with respect to Second-Order Stochastic Dominance (SOSD) in the scale of the parameters?

Let $X$ be Beta-distributed with parameters $\alpha_x,\beta_x$ and $Y$ be similarly defined. Suppose that $(\alpha_y,\beta_y)=k\cdot (\alpha_x,\beta_x)$, $k>1$. Is it the case that $Y$ second-order ...
4
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0answers
53 views

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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0answers
17 views

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 ...
5
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1answer
39 views

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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0answers
10 views

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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0answers
78 views

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 ={...
2
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1answer
98 views

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, ...
2
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0answers
24 views

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 ...
3
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1answer
37 views

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 ...
2
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0answers
23 views

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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0answers
33 views

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(...
4
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0answers
232 views

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 ...
1
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2answers
146 views

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-...
10
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1answer
352 views

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 ...
5
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1answer
99 views

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 $X\le_{\text{mrl}}Y$ ...
8
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5answers
505 views

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 ...
4
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1answer
148 views

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. ...
1
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1answer
104 views

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 ...
3
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0answers
18 views

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)...
3
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1answer
98 views

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 ...
4
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1answer
685 views

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 ...
3
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1answer
515 views

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!
3
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0answers
263 views

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 ...
5
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1answer
1k views

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 ...
4
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1answer
370 views

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\...