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Math Help - Introductory Statistics Questions

  1. #1
    lpd
    lpd is offline
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    Introductory Statistics Questions

    Hi. I really need help with these statistics problem!!

    Let X_1, ..., X_n be a random sample from a uniform distribution over (0, \theta).

    a) Find the joint density f(x_1, ... , x_n| \theta) and use the Factorization theorem to show T = max(X_1, ..., X_n) is sufficient for \theta.

    b) Find the distribution function of T and hence show the p.d.f. of T is h(t)= n(\frac{t}{n})^{n-1} \frac{1}{\theta}, 0 < t < \theta.


    Thanks for your time
    c) Find the conditional density g(x_1,... , x_n|t, \theta) and show this does not depend on \theta.

    d) Show E(T)= \frac{n}{n+1}\theta and hence find a function of the sufficient statistic that is an unbiased estimator of \theta.
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  2. #2
    MHF Contributor matheagle's Avatar
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    The joint density is

    f(x_1,\cdots , x_n)={1\over \theta ^n} I(0<X_1,\cdots ,X_n<\theta)

    where I is the indicator functions that either 0 or 1.

    NOW switch to the order stats and you can factor this as....

    f(x_1,\cdots , x_n)={1\over \theta ^n} I(0<X_{(1)})I(X_{(n)}<\theta)

    So the only stat stuck with theta is X_{(n)} our suff stat.
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