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Questions tagged [stochastic-ordering]

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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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0answers
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stochastic ordering of counting processes

Let $N_1(t)$ be a delayed renewal process and $N_2(t)$ be an ordinary renewal process such that $N_1(t)\geq_{st}N_2(t)$. Consider a renewal process $Z(t)$ with the same inter-arrivall distribution as $...
4
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1answer
58 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 $X\le_{\text{mrl}}Y$ ...
7
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3answers
191 views
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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 ...
4
votes
1answer
93 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
vote
1answer
57 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
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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)...
3
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1answer
83 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
votes
1answer
268 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
320 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
183 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
813 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
282 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\...