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Math Help - Monotone likelihood ratio

  1. #1
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    Monotone likelihood ratio

    I am trying to show that the monotone likelihood ratio for a uniform distribution [0,theta] is the maximum value (X1,...Xn). I know how to compute the monotone likelihood ratio by using fn(x given theta1)/fn(x given theta2), but then how do i show this is the maximum (X1,...Xn)?? Thanks
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  2. #2
    MHF Contributor matheagle's Avatar
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    do you know that the likelihood function is

    {1\over \theta^n} I_{X_{(1)}>0} I_{X_{(n)}<\theta}
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  3. #3
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    Yes so the MLR becomes (theta2^n)/(theta1^n). But what is the step that shows that that is the maximum (X1,...Xn)??
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  4. #4
    MHF Contributor matheagle's Avatar
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    I'm not quite sure what your two theta's are
    are you testing theta1 vs theta2?
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  5. #5
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    Monotone likelihood ratio - Wikipedia, the free encyclopedia

    It is like the formula on this page only our class uses theta1 as f(x) and theta2 as g(x)
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  6. #6
    Senior Member Sambit's Avatar
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    Suppose \theta_2>\theta_1. Then the ratio becomes \frac{\theta_1^n}{\theta_2^n}\frac{I(X_{(n)},\thet  a_2)}{I(X_{(n)},\theta_1)} [where I(a,b)=1 , if a\leq b and 0 otherwise.]

    =\frac{\theta_1^n}{\theta_2^n}\frac{1}{1} , if 0<X_{(n)}<\theta_1
    and \frac{\theta_1^n}{\theta_2^n}\frac{1}{0} if \theta_1<X_{(n)}<\infty

    that is = \frac{\theta_1^n}{\theta_2^n} ,if 0<X_{(n)}<\theta_1
    and = \infty if \theta_1<X_{(n)}<\infty

    So as X_{(n)} increases, the ratio also increases. Hence MLR in X_{(n)}.
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  7. #7
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    Thank you, but how do i show the statistic is the maximum(X1,...Xn)
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  8. #8
    Senior Member Sambit's Avatar
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    Quote Originally Posted by andyaddition View Post
    Thank you, but how do i show the statistic is the maximum(X1,...Xn)
    How do you write the pdf (or the joint pdf) of a U(0,\theta) distribution?
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  9. #9
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    1/theta^n ??
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  10. #10
    Senior Member Sambit's Avatar
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    You are missing something. The joint pdf is given by f(x_1,x_2,....,x_n)=\frac{1}{\theta^n}, if 0<X_i<\theta \forall i=1(1)n. Since all x_i are less than \theta, it is enough to ensure that max (X1,X2,....,Xn), which is denoted by X_{(n)}, is always less than \theta. Hence the joint pdf becomes f(x_1,x_2,....,x_n)=\frac{1}{\theta^n}, if 0<X_{(n)}<\theta. Now you move to the ratio.
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