# Thread: Covariance of Order Statistics from N(0,1) Distribution

1. ## Covariance of Order Statistics from N(0,1) Distribution

Hello,

I am verifing an exact value of the 2nd moment E(R^2) of a sample range (R) from the standard normal distribution. To let you clearly know what is the question that I am confronted with. Please see the attachments for details.

To save your time, you can only read the highlighted part on page 1 to get the context. My question is at the end of page 2.

Your help is highly appreciated !

Thanks a lot in advance.
DHB10 from China, PRC

2. ## Re: Covariance of Order Statistics from N(0,1) Distribution

This is page 2 of 2, my question is at the end of this page

3. ## Re: Covariance of Order Statistics from N(0,1) Distribution

Hello,

My another related question:

Because we are dealing with the random variable XY, for sure, both X and Y follow the normal distribution respectively, Is the random variable Z= XY follows the bivariate normal distribution ? If yes, we can get the joint probability density funcction f(x,y) of the random variable XY as shown in equation (3) upstairs.

Thanks a lot in advance.

4. ## Re: Covariance of Order Statistics from N(0,1) Distribution

Are you talking about X= X(1) and Y=X(n)? Because in that case X and Y don't have normal distributions.

5. ## The Following User Says Thank You to Dason For This Useful Post:

DHB10 (02-03-2012)

6. ## Re: Covariance of Order Statistics from N(0,1) Distribution

are order statistics and in general they do not follow the same distribution as the original sample. So you cannot say they have a marginal/jointly normal distribution.

To obtain the joint pdf is not hard. For example, in your case you can follow the following arguments:

To obtain , it is equivalent to:

1. Having 0 sample smaller than
2. Having 1 sample at
3. Having n - 2 sample between
4. Having 1 sample at
5. Having 0 sample larger than

And hence therefore using the multinomial arguments

7. ## The Following User Says Thank You to BGM For This Useful Post:

DHB10 (02-03-2012)

8. ## Re: Covariance of Order Statistics from N(0,1) Distribution

Hello, BGM & Dason,

Many thanks for your help.

To BGM, in your equation upstairs, are f(x) and F(x) the pdf and cdf of the standard normal distribution respectively?

Thanks
DHB10 from China, PRC

9. ## Re: Covariance of Order Statistics from N(0,1) Distribution

The equation BGM posted gives the joint distribution for the min/max of a sample of size n for any distribution where F is the cdf and f is the pdf. In your case you would want to use the pdf and cdf of the standard normal.

10. ## Re: Covariance of Order Statistics from N(0,1) Distribution

Hello, Dason,

Many thanks for your explanation!

I am going to use both your way to compute the E(R^2).

DHB10 from CN

11. ## Re: Covariance of Order Statistics from N(0,1) Distribution

@DHB10, I would suggest you look at this article:

http://www.sciencedirect.com/science...67715207002143

See Equation (12).

12. ## The Following User Says Thank You to Dragan For This Useful Post:

DHB10 (02-03-2012)

13. ## Re: Covariance of Order Statistics from N(0,1) Distribution

Hello, Dragan,

Nice to hear you again and thanks for your information, I am going to see the article you mentioned upstairs.

Thanks again.
DHB10 from China.

14. ## Re: Covariance of Order Statistics from N(0,1) Distribution

The support of the joint pdf is . So when you are integrating with this joint pdf with iterated integral, make sure that you get the integration limits of the inner integral correct - it is not integrating from to anymore.

For example, you have missed the pdf after splitting the differences?

15. ## Re: Covariance of Order Statistics from N(0,1) Distribution

Hello, BGM,

Many thanks for your help.

Meanwhile, I would like to show something interested in the following attachments. Although when n = 2, the computation result of E(R^2) is correct in my previous, at least, is the same as the author's version, however, it seems that it is coincidently correct, because according to some evidences, the probablity density function of the random variable Z = XY in this context is not what I used when computing E(R^2) in my previous post, it should have been the one shown in the following attachment ( page 3 of 3) based on the evidences shown in the following attachements.

DHB10 from China, PRC

16. ## Re: Covariance of Order Statistics from N(0,1) Distribution

This is page 2 of 3

17. ## Re: Covariance of Order Statistics from N(0,1) Distribution

You cannot substitute . is a dummy variable inside the integral which is independent of the (another dummy) variable

Also, in this example you need to ensure that you are integrating within this set: , i.e. the integration limits are in terms of

Lastly, it is not hard to show that

18. ## The Following User Says Thank You to BGM For This Useful Post:

DHB10 (02-03-2012)

19. ## Re: Covariance of Order Statistics from N(0,1) Distribution

This is page 2 of 2