Results 1 to 3 of 3

Thread: Looking for a better method (convergence in mean square)

  1. #1
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6

    Looking for a better method (convergence in mean square)

    Hi,

    I'm preparing a lesson with a student, and since he's not a major in maths, I'm looking for a better solution than mine, because it looks too "mathematical"

    We have a sample $\displaystyle M=\max(|X_1|,\dots,|X_n|)$ (iid) where Xi follows a uniform distribution over $\displaystyle [-\theta,\theta]$, where $\displaystyle \theta>0$ is a parameter.

    Previously, it was asked to find the mean and the variance of Xi. And the cdf of |Xi| (y/theta for y in [0,theta])

    Question 3) asks for the cdf of M, which is $\displaystyle G(u)=\frac{u^n}{\theta^n}$, for $\displaystyle u\in[0,\theta]$

    It's ok from here.

    Then, question 4) asks for the mean of M.
    What I did is taking the derivative of G, and then $\displaystyle \mathbb{E}(M)=\int_0^\theta u G'(u) ~du=\dots=\frac{n}{n+1}\cdot \theta$

    I'm already concerned that this is a too complicated method... So if you have a better one, denote it (1)

    Second part of question 4) asks k such that W=kM is an unbiased estimator for $\displaystyle \theta$
    Nothing magic here, $\displaystyle k=\frac{n+1}{n}$


    Third part of question 4) asks to show that W converges in mean square.
    It is not mentioned that it converges to $\displaystyle \theta$. So how can I explain to the boy that it should converge to theta ?
    This is the most awful part, because what I did is to use the property that :
    $\displaystyle W$ converges to a constant c $\displaystyle \Longleftrightarrow$ $\displaystyle \lim_{n\to\infty}\mathbb{E}(W)=\theta$ and $\displaystyle \lim_{n\to\infty} \mathbb{V}\text{ar}(W)=0$

    But calculating the variance of W is a really ugly... So if you have a better solution, please denote it (2)

    And if you think these are the only ways to solve the questions, just eat a T-bone steak for dinner


    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    I'm not exactly sure what your asking, and I'm almost certainly sure you'll yell at me.

    BUT I would ignore chebyshev's and just show that $\displaystyle W_n\to\theta$ as n goes to infinity

    That is, just use the definition $\displaystyle E|W_n-\theta|^2\to 0$

    And you do need to calculate the moments, but instead of the variance I would just use

    $\displaystyle E|W_n-\theta|^2=E(W^2_n)-2\theta E(W_n)+\theta^2$

    and show that this approaches zero as n goes to infinity.
    Last edited by matheagle; Aug 13th 2009 at 12:22 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Thanks, it simplifies a bit to use E|W-theta|
    And no, I won't yell
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Solve Using the Square Root Method
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Dec 11th 2010, 07:42 PM
  2. [SOLVED] Help for Completing Square method
    Posted in the Algebra Forum
    Replies: 4
    Last Post: Jul 10th 2010, 06:28 AM
  3. least square method got stuck
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Jan 27th 2010, 04:42 PM
  4. Square Root Method
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Oct 25th 2009, 02:14 PM
  5. graeco latin square method
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: Nov 7th 2007, 02:59 PM

Search Tags


/mathhelpforum @mathhelpforum