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

#### DHB10

##### New Member
Hello, BGM,

Many thanks for your great help! So, this topic can be closed.

Thanks again !
DHB10

#### DHB10

##### New Member
Hello, BGM,

Would you please to advise how to transfer the original integral to be a gamma integral in the case of n= 3 ?

Thanks

DHB10

#### DHB10

##### New Member
Hello,

Please help with the trick from row 1 to row 2 ? Pls see the attached below.

Thanks

#### BGM

##### TS Contributor
The trick is look at $$-\infty < x < t < y < +\infty$$

So if the order is $$x, y, t$$, then you will say

$$-\infty < x < t, t < y < +\infty, -\infty < t < +\infty$$

while if the order is $$t, y, x$$, then you will say

$$x < t < y, x < y < +\infty, -\infty < x < +\infty$$

So you will choose the "tightest" bound, in terms of the variable that not integrated out yet.

You can choose any order freely and try.

Of course it has to satisfy the condition of Fubini's Theorem.

#### DHB10

##### New Member
Hello, BGM,

Thanks a lot. Very interesting !

#### DHB10

##### New Member
Hello, BGM,

Fubini's theorem tells us that if the integral of the absolute value is finite, then the order of integration does not matter; if we integrate first with respect to x and then with respect to y, we get the same result as if we integrate first with respect to y and then with respect to x.

Question: What a person is doing is to get the value of an iterated integral, how does s/he know the integral of the absolute value is finite or infinite in advance?

Thanks

#### DHB10

##### New Member
Hello, BGM,

One more question, I would like to compute E(R^2) when n=5 using the approach in your previous post, I need to confirm a joint pdf, which is based on your example posted for the case of n =4. Please see the attached below for a simple question to you.

Thanks a lot.
DHB10

#### BGM

##### TS Contributor
Yes, because $$n!$$ is the normalizing constant accounts for the permutation.

For $$n = 5$$ or above it will be quite different, because you will end up with the integral like

$$E[X_{(1)}X_{(n)}] = -\frac {n!} {(2\pi)^{\frac {n} {2}}}$$$$\idotsint\limits_{-\infty < x_2 < x_3 < ... < x_{n-1} < +\infty} \exp\left\{-x_2^2-x_{n-1}^2 - \frac {1} {2} \sum_{i=3}^{n-2} x_i^2\right\} dx_2dx_3\cdots dx_{n-1}$$

which again need to find the volume of the unit (n-3)-ball intersect with the support, by the (hyper)spherical coordinates.

#### DHB10

##### New Member
Hello,

Yes, exactly, while I am trying the case of n = 5, there are two integrations bounds in spherical coordinates, which have to be correctly determined. it is hard to correctly get the bounds. Maybe, it is quite easy for you. Please help with this bounds. Please see below attachment for more.

Thanks a lot!

DHB10

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#### DHB10

##### New Member
Hello,

For case of n = 5, it seems that changing to spherical coordinates cannot work because the bounds cannot be defined. Someone in the world had solved this case, because the exact value of E(R^2),when n = 5, had been computed exactly as shown in the table in my first post. So,we need to find out another trick to handle the integral. Please help with this.

Thanks

#### DHB10

##### New Member
Hello, BGM,

Do you have any trick to handle this integral in the sphrecial coordinates ? or any alternative approach instead of changing to sphericial coordinates?

Thanks

#### BGM

##### TS Contributor
Yes they are equivalent and can be proved.

#### DHB10

##### New Member
Hello, BGM,

Many thanks for your help. So, when n = 5, with your confirmation upstairs, my computation formula is correct now. I have checked the computation many times by now, It seems that there is no mistake included. I am curious why the final value is incorrect now that there is no mistake. Maybe it is easier for you to detect the mistake somewhere. It couldn't be better anymore if you can find out the mistake in the computation upstairs before closing this topic.

Thanks a lot.
DHB10

#### DHB10

##### New Member
Hello,

This topic is approaching to the end. Only the last exact value verification is pending now. I cannot do a simulation to verify the data, need your help.

Thanks
DHB10

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#### BGM

##### TS Contributor
Actually I have already verified the posted answer is correct before, but have not post it here

I have not seriously check your steps but it looks good to me.

#### DHB10

##### New Member
Hello, Dragan,

It's my pleasure to hear from you again. As I said in my previous post, regarding the integral bounds of phi, I don't know how to derive the bounds of phi mathematically. I just tried out several bounds of phi, finding out that only this one can work. So, it couldn't be better anymore if you would like to take a little time to give me some trick or guidance to mathematically prove or derive this integral bounds of phi.

Many thanks for the help from Dragan and BGM.

DHB10

#### DHB10

##### New Member
Hello, Dragan & BGM,

With the great help from both of you, I have finished the computation. Meanwhile, I have already mathematically graphically proved the integral bounds of phi. To close this topic, I would like to share my revised version with those who might meet the same issue or who are concerned about this topic.

It couldn't be better anymore if you would like to advise me the mathematically rigorous proof for the integral bounds of phi.

Many thanks for your past help in the past few weeks. I am going to focus my energy and time on other job from tomorrow onwards. I will visit this excellent forum from time to time in the future.

Best Regards
DHB10 from China, PRC

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#### DHB10

##### New Member
This is page 2 of 2

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#### DHB10

##### New Member
Hello, BGM,

Is the graphic proof of the integral bounds of phi in my previous post correct ? I was trying to find out an alternative way to prove this bounds. but I have not got one. Would you please to give me some hint or tip ?

Thanks a lot