Results 1 to 10 of 10

Thread: Monotone likelihood ratio

  1. #1
    Junior Member
    Joined
    Dec 2008
    Posts
    37

    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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    do you know that the likelihood function is

    $\displaystyle {1\over \theta^n} I_{X_{(1)}>0} I_{X_{(n)}<\theta}$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2008
    Posts
    37
    Yes so the MLR becomes (theta2^n)/(theta1^n). But what is the step that shows that that is the maximum (X1,...Xn)??
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    I'm not quite sure what your two theta's are
    are you testing theta1 vs theta2?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Dec 2008
    Posts
    37
    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)
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member Sambit's Avatar
    Joined
    Oct 2010
    Posts
    355
    Suppose $\displaystyle \theta_2>\theta_1$. Then the ratio becomes $\displaystyle \frac{\theta_1^n}{\theta_2^n}\frac{I(X_{(n)},\thet a_2)}{I(X_{(n)},\theta_1)}$ [where $\displaystyle I(a,b)=1 ,$ if $\displaystyle a\leq b$ and 0 otherwise.]

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

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

    So as $\displaystyle X_{(n)}$ increases, the ratio also increases. Hence MLR in $\displaystyle X_{(n)}$.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Dec 2008
    Posts
    37
    Thank you, but how do i show the statistic is the maximum(X1,...Xn)
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member Sambit's Avatar
    Joined
    Oct 2010
    Posts
    355
    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 $\displaystyle U(0,\theta)$ distribution?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Dec 2008
    Posts
    37
    1/theta^n ??
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Senior Member Sambit's Avatar
    Joined
    Oct 2010
    Posts
    355
    You are missing something. The joint pdf is given by $\displaystyle f(x_1,x_2,....,x_n)=\frac{1}{\theta^n}, $ if $\displaystyle 0<X_i<\theta $ $\displaystyle \forall i=1(1)n$. Since all $\displaystyle x_i$ are less than $\displaystyle \theta$, it is enough to ensure that max (X1,X2,....,Xn), which is denoted by $\displaystyle X_{(n)}$, is always less than $\displaystyle \theta$. Hence the joint pdf becomes $\displaystyle f(x_1,x_2,....,x_n)=\frac{1}{\theta^n}, $ if $\displaystyle 0<X_{(n)}<\theta$. Now you move to the ratio.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. monotone likelihood ratio
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Apr 26th 2011, 11:02 PM
  2. likelihood ratio test
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: Aug 6th 2010, 01:04 PM
  3. generalized likelihood ratio
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Feb 16th 2010, 07:18 PM
  4. Likelihood Ratio Test & Likelihood Function
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: Apr 23rd 2009, 06:53 AM
  5. Likelihood Ratio Test
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: Apr 8th 2009, 09:33 PM

Search Tags


/mathhelpforum @mathhelpforum