Submission declined on 21 March 2023 by
Newystats (
talk).
Where to get help
How to improve a draft
You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Editor resources
|
This article generalize the space of random variables generated by exponential Orlicz function, which in turn can be regarded as generalization of space for random variables. The sub-exponential distribution discussed here is heavily related to sub-Gaussian distribution. Do not be confused with the sub-exponential distribution in heavy-tailed distribution.
In probability theory, a sub-exponential distribution is a probability distribution with exponential tail decay. Informally, the tails of a sub-exponential distribution decay at a rate similar to those of the tails of a exponential random variable. This property gives sub-exponential distributions their name.
Formally, the probability distribution of a random variable is called sub-exponential if there are positive constant C such that for every ,
The sub-exponential distribution is heavily related to sub-Gaussian distribution. In fact, the square of a sub-exponential is sub-Gaussian [1], which has an even stronger tail decay.
Let be a random variable. The following conditions are equivalent:
Proof. By the layer cake representation,After a change of variables , we find that
Using the Taylor series for :and monotone convergence theorem, we obtain thatwhich is less than or equal to for . Take , then
A random variable is called a sub-exponential random variable if either one of the equivalent conditions above holds.
The sub-exponential norm of , denoted as , is defined bywhich is the Orlicz norm of generated by the Orlicz function By condition above, sub-exponential random variables can be characterized as those random variables with finite sub-exponential norm.
A random variable is sub-Gaussian if and only if is sub-exponential. Moreover, .
Proof. This follows easily from the characterization of the random variables by the sub-exponential norm and sub-Gaussian norm. Indeed, by definition,Hence, we find that . Therefore, one of the norm is finite if and only if another one is. This shows that is sub-Gaussian if and only if is sub-exponential.
The following properties are equivalent:
If has exponential distribution with rate , i.e. , thenThen is sub-exponential since it satisfy condition with .
Submission declined on 21 March 2023 by
Newystats (
talk). This draft's references do not show that the subject
qualifies for a Wikipedia article. In summary, the draft needs multiple published sources that are:
Where to get help
How to improve a draft
You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Editor resources
|
This article generalize the space of random variables generated by exponential Orlicz function, which in turn can be regarded as generalization of space for random variables. The sub-exponential distribution discussed here is heavily related to sub-Gaussian distribution. Do not be confused with the sub-exponential distribution in heavy-tailed distribution.
In probability theory, a sub-exponential distribution is a probability distribution with exponential tail decay. Informally, the tails of a sub-exponential distribution decay at a rate similar to those of the tails of a exponential random variable. This property gives sub-exponential distributions their name.
Formally, the probability distribution of a random variable is called sub-exponential if there are positive constant C such that for every ,
The sub-exponential distribution is heavily related to sub-Gaussian distribution. In fact, the square of a sub-exponential is sub-Gaussian [1], which has an even stronger tail decay.
Let be a random variable. The following conditions are equivalent:
Proof. By the layer cake representation,After a change of variables , we find that
Using the Taylor series for :and monotone convergence theorem, we obtain thatwhich is less than or equal to for . Take , then
A random variable is called a sub-exponential random variable if either one of the equivalent conditions above holds.
The sub-exponential norm of , denoted as , is defined bywhich is the Orlicz norm of generated by the Orlicz function By condition above, sub-exponential random variables can be characterized as those random variables with finite sub-exponential norm.
A random variable is sub-Gaussian if and only if is sub-exponential. Moreover, .
Proof. This follows easily from the characterization of the random variables by the sub-exponential norm and sub-Gaussian norm. Indeed, by definition,Hence, we find that . Therefore, one of the norm is finite if and only if another one is. This shows that is sub-Gaussian if and only if is sub-exponential.
The following properties are equivalent:
If has exponential distribution with rate , i.e. , thenThen is sub-exponential since it satisfy condition with .
-
in-depth (not just passing mentions about the subject)
-
reliable
-
secondary
-
independent of the subject
Make sure you add references that meet these criteria before resubmitting. Learn about mistakes to avoid when addressing this issue. If no additional references exist, the subject is not suitable for Wikipedia.