Skip to main content

Questions tagged [stochastic-ordering]

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

Filter by
Sorted by
Tagged with
0 votes
1 answer
34 views

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 ...
dodo's user avatar
  • 185
8 votes
1 answer
141 views

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 ...
jginestet's user avatar
  • 1,994
2 votes
0 answers
98 views

Projection of i.i.d. Gaussians onto correlated Gaussian is bounded from below by chi-squared

Let $\varepsilon \sim \mathcal{N}(0, \mathrm{Id}_k)$ and $\varepsilon_0 \in \mathbb{R}$ with $\varepsilon_0 \neq 0$. For some matrix or vector $A \in \mathbb{R}^{k \times p}$, let $P_A := A (A^T A)^{-...
M. Londschien's user avatar
0 votes
0 answers
15 views

Does the definition of a mean-preserving spread not imply increased variance in an underlying distribution?

A random variable y is a mean preserving spread of the random variable x iff y = x + e where e is noise and E(e|x)=0 (per notes here). And yet, two random variables possessing equal means and ...
ordering-lotteries-help's user avatar
4 votes
1 answer
143 views

Is the relation "not FOSD" transitive?

$X:\Omega\to [0,1]$ is a random variable. It is known that first order stochastic dominance FOSD is a partial order that is transitive: $X$ FOSD $Y$, $Y$ FOSD $Z$ implies $X$ FOSD $Z$. Now consider ...
High GPA's user avatar
  • 823
0 votes
0 answers
29 views

Predicting willingness-to-pay for a risk-averse person who can 'select' lotteries

I'm studying how the willingness-to-pay differs for individuals who can 'select' lotteries. Individuals are presented with L1 first and can pay some amount to get lottery L2. Assume these are my ...
ordering-lotteries-help's user avatar
0 votes
0 answers
26 views

Question about first order stochastic dominance

Suppose $x \sim F(x; s)$ where $s\in\{n, p\}$. Then let $F(x; n) \leq F(x; p) \forall x$. Can this be be a valid PDF? It seems that other than the case where $F(x;n)=F(x;p)$ it is not possible. ...
L1234's user avatar
  • 1
1 vote
1 answer
72 views

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....
daruma's user avatar
  • 217
1 vote
0 answers
44 views

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 (...
Galen's user avatar
  • 9,412
4 votes
2 answers
88 views

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 ...
sczinner's user avatar
1 vote
0 answers
56 views

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 ...
dodo's user avatar
  • 185
6 votes
2 answers
469 views

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 ...
Yashaswi Mohanty's user avatar
8 votes
4 answers
2k views

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 ...
Galen's user avatar
  • 9,412
0 votes
0 answers
25 views

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 $...
Jed's user avatar
  • 61
2 votes
0 answers
238 views

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 ...
oyy's user avatar
  • 65
2 votes
1 answer
127 views

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 ...
EconJohn's user avatar
  • 892
0 votes
0 answers
151 views

"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/...
kpr62's user avatar
  • 101
2 votes
0 answers
49 views

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 ...
ChooCheeDuck's user avatar
3 votes
1 answer
113 views

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 ...
oyy's user avatar
  • 65
1 vote
0 answers
115 views

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 ...
RGG's user avatar
  • 73
1 vote
0 answers
32 views

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 ...
GabMac's user avatar
  • 267
3 votes
1 answer
581 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 ...
Dayne's user avatar
  • 2,661
5 votes
0 answers
419 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 ...
doubled's user avatar
  • 4,977
1 vote
0 answers
23 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 ...
Saar's user avatar
  • 11
7 votes
1 answer
584 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 ...
Julian Karch's user avatar
  • 1,970
2 votes
0 answers
899 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 ={...
Painkiller's user avatar
2 votes
1 answer
455 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, ...
Alex 's user avatar
  • 43
2 votes
0 answers
48 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 ...
minc33's user avatar
  • 83
4 votes
1 answer
54 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 ...
andrewH's user avatar
  • 3,157
2 votes
0 answers
28 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 $B$?
robust's user avatar
  • 265
1 vote
1 answer
152 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(...
CVCM's user avatar
  • 11
13 votes
0 answers
2k 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 ...
dlaehnemann's user avatar
1 vote
2 answers
516 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-...
Paul's user avatar
  • 33
10 votes
1 answer
484 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 ...
Sextus Empiricus's user avatar
5 votes
1 answer
173 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 $X\le_{\text{mrl}}Y$ ...
Jimmy R.'s user avatar
  • 214
12 votes
5 answers
1k 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 ...
innisfree's user avatar
  • 1,540
4 votes
1 answer
696 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. ...
Green.H's user avatar
  • 141
1 vote
1 answer
374 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 ...
Stochastic's user avatar
4 votes
0 answers
22 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)...
Sus20200's user avatar
  • 381
4 votes
1 answer
106 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 ...
Sus20200's user avatar
  • 381
5 votes
1 answer
1k 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 ...
John's user avatar
  • 435
4 votes
1 answer
870 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!
Hirek's user avatar
  • 1,007
5 votes
0 answers
378 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 ...
Hirek's user avatar
  • 1,007
5 votes
1 answer
2k 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 ...
bonna's user avatar
  • 143
4 votes
1 answer
664 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\...
firemind's user avatar
  • 233