Results 1 to 6 of 6

Math Help - Point Estimates

  1. #1
    Junior Member
    Joined
    Nov 2009
    From
    Australia
    Posts
    56

    Point Estimates

    Point Estimate Question

    Hi,
    May I confirm if my answers to the questions are correct?

    For Q1) I used the following limits for my integral
    \int_{x_{0}}^{0} \alpha x_{0}^{\alpha}\ x^{-\alpha-1}\ dx = 1

    Q2) My final answer is:
    \alpha = \frac{\mu_{1}}{\mu_{1} - x_{0}}

    Q3) My answer after solving the derivative of the log likelihood
    \alpha = \frac{1}{\log \left( \frac{x}{x_{0}}   \right)}

    Thank you in advance
    Linda
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5

    Re: Point Estimates

    the upper bound should be infinity not zero
    I get a slightly differenet answer for part 2
    and for the MLE, do we have a sample or just one observation?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2009
    From
    Australia
    Posts
    56

    Re: Point Estimates

    Hi Matt,

    Thank you for pointing out that the limit bound should be infinity!

    May I ask what you got for part 2?
    They way I arrived at my answer was I found the expectation of the distribution first:
    E[X] = \alpha x_{0}^{\alpha} \int_{x_{0}}^{\infty} x \times x^{-\alpha-1}\ dx
    =\frac{\alpha x_{0}}{\alpha -1}

    I then substituted the E[X] with the first moment \mu_{1} and then rearranged the above term to obtain \alpha

    w.r.t MLE, that was how the question was given, we had no further data to work with.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5

    Re: Point Estimates

    it's math eagle
    the mean is correct, next you solve for alpha

    \frac{\alpha x_{0}}{\alpha -1}=\bar X

    \frac{\alpha -1}{\alpha}=  {x_0\over \bar X}

    1-\frac{1}{\alpha}= {x_0\over \bar X}

    ok, now you finish
    Last edited by matheagle; June 21st 2011 at 07:59 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Nov 2009
    From
    Australia
    Posts
    56

    Re: Point Estimates

    Quote Originally Posted by matheagle View Post
    it's math eagle
    Oh no! I am so sorry about getting you're alias incorrect. Forgive me for my dumbness.
    I know someone with the surname Heagle, and so originally read it as Mat Heagle. My apologies!!!


    1-\frac{1}{\alpha}= {x_0\over \bar X}

    ok, now you finish
     \\ -\frac{1}{\alpha} = \frac{x_{0}}{\bar{X}} - 1 \\  \frac{1}{\alpha} = 1 - \frac{x_{0}}{\bar{X}} \\  \frac{1}{1 - \frac{x_{0}}{\bar{X}}} = \alpha \\  \alpha = \frac{\bar{X}}{\bar{X}-x_{0}}

    I see that I should have substituted \bar{X} for \mu_{1}. Thank you!!

    As a further I have updated my answer for the MLE to:
    \alpha = \frac{n}{\sum\log \left( \frac{x_{i}}{x_{0}}   \right)}
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5

    Re: Point Estimates

    its ok, I've been called much worse, its just not matt
    again with the MLE, usually you have a sample of i.i.d. rvs.
    But sometimes it's just one observation.
    It isn't clear, but most likely its the sample of n observations.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. estimates
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: March 9th 2011, 10:57 PM
  2. Interval estimates?
    Posted in the Statistics Forum
    Replies: 3
    Last Post: May 11th 2010, 09:11 PM
  3. Interval Estimates
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: June 27th 2009, 12:04 AM
  4. point estimates and endpoints of confidence interval
    Posted in the Advanced Statistics Forum
    Replies: 12
    Last Post: June 13th 2009, 09:55 PM
  5. Unbiased Estimates
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 22nd 2008, 12:15 AM

Search Tags


/mathhelpforum @mathhelpforum