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Thread: Distribution of the Sample Range

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    Distribution of the Sample Range




    Hello,

    I got an article that was written by an unknown author, which I downloded from the Internet. It looks like one part of a book. I have two questions about the distribution of the sample range. My questions are included in the attachment.

    Please help with the said two questions

    Thanks a lot in advance.

    DHB10 from China, PRC
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    Last edited by DHB10; 01-30-2012 at 12:31 PM.

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    Re: Distribution of the Sample Range

    My Question 2
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    Re: Distribution of the Sample Range

    Without doing any integration, I have use simple simulation to verify that the answer 1 + \frac {\sqrt{3}} {\pi} should be correct.

    For the question 2, please note that X_{(n)}, X_{(1)} are not independent. And thus E[X_{(n)}X_{(1)}] \neq E[X_{(n)}]E[X_{(1)}] in general.

    To integrate the cross moments you will need to use the joint pdf.

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    Re: Distribution of the Sample Range

    For your first question: Simulating this seems to suggest the author's version is correct.

    For the second: E[XY] = E[X]E[Y] only works in general for independent random variables. X(1) and X(n) are not independent.

    Edit: Wow - BGM ninja'd me and we said essentially the same thing!

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    Re: Distribution of the Sample Range

    Hello,

    The following two pages show my computation process about the said question in my post upstairs . My computation is based on the formula in the following article.

    R. C. Bose and Shanti S. Gupte, Moments of Order Statistics from a Normal Population, Biometrika 46(1959), 433-440.

    My computation is different from the said author in this topic. I double checked my computation, I failed to find out any mistake. So, I am wondering whether or not the answer of E(X^2(4,4)) shown in the table is correct. Please help with this question. Thanks.
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    Last edited by DHB10; 01-30-2012 at 02:38 PM.

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    Re: Distribution of the Sample Range

    This is the second page of my compuation.
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    Last edited by DHB10; 01-30-2012 at 02:42 PM.

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    Re: Distribution of the Sample Range

    Hello, BGM,

    Many thanks for your fast reply and help. Would you please to verify your conclusion by integration ?

    Thanks again.
    DHB10 from China, PRC
    Last edited by DHB10; 01-30-2012 at 01:36 PM.

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    Re: Distribution of the Sample Range

    Hello, Bason,

    Thanks for your reply and the help. Would you please to verify the author's version by integration ?

    Thanks in advance.
    DHB10 from China, PRC

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    Re: Distribution of the Sample Range

    Nope. Too much work - that's for you to do. The simulation shows that the author is correct so something went wrong with your integration. Just a quick question: You mention near the end of your proof that the integrand is odd with respect to x - which is fine. But you were integrating that quantity with respect to theta. I'm not sure if that makes a difference here or not but that might be one possible place where you're getting messed up here.

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    Re: Distribution of the Sample Range

    Hello,

    Question 3

    For the same page, by my application to a Xbar-R chart, I find that when n=4, it seems that the author's E(R^2) is not correct in the table. Please see the attached below. I tried my tentative answer, it should be corret one. but I have not yet carried out a rigorous mathematical proof to prove my tentative answer. Please help with this.

    Thanks in advance
    DHB10 from China, PRC
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    Last edited by DHB10; 01-30-2012 at 12:46 PM.

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    Re: Distribution of the Sample Range

    Simulation suggests that the author is incorrect on that one and your answer is correct. Have you ever used R? You could check these yourself through simulation quite easily to see if you're correct or if the author is correct (or neither?)

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    Re: Distribution of the Sample Range

    Hello, Dason,

    Yes, I have used R, which was why I found that the author's E(R^2) is not correct when n=4.
    What software are you using to simulate ?

    thanks
    DHB10 from China, PRC

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    Re: Distribution of the Sample Range

    The one I asked you about: R.

    To do the last simulation it was as easy as
    Code: 
    
    > j <- replicate(500000, diff(range(rnorm(4)))^2)
    > mean(j)
    [1] 5.010641
    > 2 + (6+2*sqrt(3))/pi
    [1] 5.012517
    > 2 + (6+3*sqrt(3))/pi
    [1] 5.563846

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    Re: Distribution of the Sample Range

    Hello, Dason,

    Yes, I did make a mistyping, should be integration with respect to x first, I updated my attachement ( page 2 of 2) , please see the following attachment.

    thanks
    DHB10
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    Re: Distribution of the Sample Range


    Hello, Global Moderators or Statistics Experts,

    I double checked my integration shown upstairs in this topic , I still failed to detect any mistake or nonconformance to calculus theory. However, the simulation of both BGM and Dason suggest that the author's version be correct (E(X^2(4,4)). My integration result is different from the author version in terms of E(X^2(4,4)). So, what is the problem ? If my version is incorrect, where is the problem located in the integration ? Please help with this.

    Thanks a lot in advance.
    DHB10 from China, PRC

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