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

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    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
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    Last edited by DHB10; 02-03-2012 at 01:22 AM.

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    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
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    Last edited by DHB10; 02-03-2012 at 12:31 AM.

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    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.
    Last edited by DHB10; 02-03-2012 at 01:30 AM.

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    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.

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

    X_{(1)}, X_{(n)} 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 X_{(1)} = x_1, X_{(n)} = x_n, it is equivalent to:

    1. Having 0 sample smaller than x_1
    2. Having 1 sample at x_1
    3. Having n - 2 sample between x_1, x_n
    4. Having 1 sample at x_n
    5. Having 0 sample larger than x_n

    And hence therefore using the multinomial arguments
    f_{X_{(1)}, X_{(n)}}(x_1, x_n)dx_1dx_n = \frac {n!} {0!1!(n-2)!1!0!} F(x_1)^0 f(x_1)dx_1 [F(x_n) - F(x_1)]^{n-2} f(x_n)dx_n [1 - F(x_n)]^0

    \Rightarrow  f_{X_{(1)}, X_{(n)}}(x_1, x_n) = n(n - 1)[F(x_n) - F(x_1)]^{n-2}f(x_1)f(x_n), -\infty < x_1 < x_n < +\infty

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    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

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    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.

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    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

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    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).

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    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.

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

    The support of the joint pdf is -\infty < x_1 < x_n < +\infty. 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 - \infty to + \infty anymore.

    For n = 3 example, you have missed the pdf after splitting the differences?

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

    This is page 2 of 3
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    Last edited by DHB10; 02-03-2012 at 03:50 PM.

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

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

    Also, in this example you need to ensure that you are integrating within this set: -\infty < x < \frac {z} {x} < +\infty, i.e. the integration limits are in terms of z

    Lastly, it is not hard to show that

    \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} g(x,y)f_{X,Y}(x,y)dxdy = E[g(X, Y)] = \int_{-\infty}^{+\infty} z f_{g(X,Y)}(z)dz

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


    This is page 2 of 2
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    Last edited by DHB10; 02-03-2012 at 09:35 AM.

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