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Thread: can we use bassu theorem?

  1. #1
    alv
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    Question can we use bassu theorem?

    Suppose X1,X2,X3,... are a sequence of independent random variables with
    uniform distribution on (0, 1). If N = min{n>0 | Xn:n - X1:n>a, 0<a<1 and a is constant}.
    Xn:n = max Xi, 1 =< i =<n
    X1:n = min Xi, 1 =< i =<n.
    What is E[N]?

    ps. 1)does that sort of stating the problem implies basu theorem {theorem stating the independence of a complete sufficient statistic and an ancillary statistic}or not?
    ps. 2)n:n and 1:n are index.
    Last edited by alv; Jul 13th 2008 at 09:50 PM. Reason: font change
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  2. #2
    Flow Master
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    Quote Originally Posted by alv View Post
    Suppose X1,X2,X3,... are a sequence of independent random variables with






    uniform distribution on (0, 1). If N = min{n>0 | Xn:n - X1:n>a, 0<a<1 and a is constant}.
    Xn:n = max Xi, 1 =< i =<n
    X1:n = min Xi, 1 =< i =<n.
    What is E[N]?

    ps. 1)does that sort of stating the problem implies basu theorem {theorem stating the independence of a complete sufficient statistic and an ancillary statistic}or not?

    ps. 2)n:n and 1:n are index.
    Xn:n and X1:n are random variables so I'm not quite sure how to interpret the statement Xn:n - X1:n>a ....? Do you mean that Pr(Xn:n - X1:n>a) is greater than some value ....?

    Be that as it may, the following might help ...... I'm changing the notation to represent Xn:n by $\displaystyle X_{(n)}$ and X1:n by $\displaystyle X_{(1)}$.


    Let $\displaystyle X_1, \, X_2, \, .... \, X_n$ be a sequence of i.i.d. random variables with pdf denoted by f(x), cdf denoted by F(x) and order statistics $\displaystyle X_{(1)}, \, X_{(2)}, \, .... \, X_{(n)}$.

    It can be shown that the pdf of the interval $\displaystyle W_{rs} = X_{(s)} - X_{(r)}$ (where $\displaystyle 1 \leq r < s \leq n$ ) is given by


    $\displaystyle f(w_{rs}) = \frac{n!}{(r-1)! (s-r-1)! (n-s)!}$ $\displaystyle \int_{-\infty}^{+\infty} [ F(x) ]^{r-1} f(x) \, [ F(x + w_{rs}) - F(x) ]^{s-r-1} \, f(x + w_{rs}) \, [1 - F(x + w_{rs}) ]^{n-s} \, dx$ .... (1).


    When r = 1 and s = n the interval $\displaystyle W_{rs}$ becomes the range W and equation (1) reduces to:


    $\displaystyle f(w) = n (n-1) \, \int_{-\infty}^{+\infty} f(x) \, [ F(x + w) - F(x) ]^{n-2} \, f(x + w) \, dx$ .... (2).


    When $\displaystyle X_i$ ~ $\displaystyle U(0, 1)$, f(x) = 1 for $\displaystyle 0 \leq x \leq 1$ and zero elsewhere and equation (2) becomes:


    $\displaystyle f(w) = n (n-1) \, \int_{0}^{1-w} (1) \, [ x + w - x ]^{n-2} \, (1) \, dx = n (n-1) \, \int_{0}^{1-w} w^{n-2} dx$

    where the upper integral terminal is because f(x + w) = 0 for $\displaystyle x \geq 1 - w$. Therefore the pdf of the range of n numbers selected at random from the interval (0, 1) is:


    $\displaystyle f(w) = n(n-1)w^{n-2} (1 - w) ~ , 0 \leq w \leq 1$ .... (3).


    Note by the way the interesting result that $\displaystyle E(W) = \frac{n-1}{n+1}$ and E(W) = 1 in the limit n --> oo ....
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