Issue in understanding stochastic ordering

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In summary, stochastic ordering is a concept used to compare two random variables based on their respective probabilities. The rule states that one random variable is less than or equal to another if the expectation of a non-decreasing function is smaller or equal. This rule holds for all non-decreasing functions, even for the special case where the function is defined as 0 for values less than or equal to x and 1 for values greater than x. This hypothesis can be extended to all non-decreasing functions through a limiting argument.
  • #1
pakkanen
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This is not actually a problem but I need clarification for stochastic ordering

From Wikipedia, there is stated: (http://en.wikipedia.org/wiki/Stochastic_ordering)

Usual stochastic order:
a real random variable A is less than a random variable B in the "usual stochastic order" if
Pr(A>x) ≤ Pr(B>x) ,where Pr(°) denotes the probability of an event.

Here comes the issue that I do not understand:
Characterizations:
The following rules describe cases when one random variable is stochastically less than or equal to another. Strict version of some of these rules also exist.

1. A ≤ B if and only if for all non-decreasing functions u, E[u(A)] ≤ E[u(B)].

Why it is not possible to attain Pr(A>x) ≥ Pr(B>x) for some x even if E[u(A)] ≤ E[u(B)] ?
 
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  • #2
pakkanen said:
This is not actually a problem but I need clarification for stochastic ordering

From Wikipedia, there is stated: (http://en.wikipedia.org/wiki/Stochastic_ordering)

Usual stochastic order:
a real random variable A is less than a random variable B in the "usual stochastic order" if
Pr(A>x) ≤ Pr(B>x) ,where Pr(°) denotes the probability of an event.

Here comes the issue that I do not understand:
Characterizations:
The following rules describe cases when one random variable is stochastically less than or equal to another. Strict version of some of these rules also exist.

1. A ≤ B if and only if for all non-decreasing functions u, E[u(A)] ≤ E[u(B)].

Why it is not possible to attain Pr(A>x) ≥ Pr(B>x) for some x even if E[u(A)] ≤ E[u(B)] ?


The inequality E[u(A)] ≤ E[u(B)] is assumed to hold for ALL non-decreasing functions. For the special case where u(t) = 0 if t ≤ x and u(t) = 1 if t > x (a non-decreasing function) we have E[u(A)] = Pr(A>x), etc.

Note: we can assume the seemingly-weaker hypothesis that E[u(A)] ≤ E[u(B)] holds for all strictly increasing u; then, by a limiting argument we can prove it for all non-decreasing u. Therefore, the special u above is in no sense "artificial".

RGV
 
  • #3
Late gratitudes! I believe I get it.
 

1. What is stochastic ordering?

Stochastic ordering is a mathematical concept that is used to compare and rank random variables in terms of their probability distributions. It helps us understand the relationship between different variables in terms of their likelihood of occurrence.

2. How is stochastic ordering useful in scientific research?

Stochastic ordering allows scientists to analyze and interpret data in a systematic and quantitative manner. It can be applied to various fields such as economics, biology, and engineering, to name a few. It helps in drawing conclusions and making predictions based on statistical evidence.

3. What are the different types of stochastic ordering?

There are three main types of stochastic ordering: first-order, second-order, and third-order. First-order stochastic ordering compares the means of two distributions, second-order compares the variances, and third-order compares the skewness.

4. How do you establish stochastic ordering?

Stochastic ordering can be established through various methods such as graphical analysis, mathematical proofs, and statistical tests. The choice of method depends on the data and the research question being addressed.

5. What are some common challenges in understanding stochastic ordering?

One of the main challenges in understanding stochastic ordering is the complexity of the mathematical concepts involved. It also requires a strong foundation in statistics and probability theory. Additionally, interpreting the results of stochastic ordering can be difficult, as it is a comparative rather than absolute measure.

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