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Math Help - Unbiased Estimation and Method of Moments

  1. #1
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    Unbiased Estimation and Method of Moments

    I need your help...Please read the document that I attached here...thanks!
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    Last edited by mr fantastic; January 10th 2009 at 08:08 PM. Reason: Removed erroneous effort at attachment
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    To save others the trouble of opening the attachment:

    1. Let x_1, \, x_2, \, .... , \, x_n be a sample from a Bernoulli distribution with parameter p.

    P[X = x] = p^x (1 - p)^{1-x}, \, I_{(0, 1)}(x)

    a. Derive the method of moments estimator of p.

    b. Verify if your method of moments estimator of p is unbiased for p.


    2. Let x_1, \, x_2, \, .... , \, x_n be a sample from a Gamma distribution with parameter \alpha and \beta.

    f(x) = \frac{x^{\alpha - 1} e^{-x/\beta}}{\Gamma(\alpha) \beta^{\alpha}}, \, x > 0, \, \beta > 0

    f(x) = 0, ~ x \leq 0

    a. If \beta is known, derive the method of moments estimator of \alpha.

    b. Verify if your method of moments estimator of \alpha is unbiased for \alpha.


    Quote Originally Posted by aadbaluyot View Post
    I need your help...Please read the document that I attached here...thanks!
    First read these threads:

    http://www.mathhelpforum.com/math-he...tion-help.html (posts #1, #2)

    http://www.mathhelpforum.com/math-he...tatistics.html (posts #1, #2)

    http://www.mathhelpforum.com/math-he...estimator.html

    http://www.mathhelpforum.com/math-he...estimator.html

    http://www.mathhelpforum.com/math-he...estimator.html


    1. a. E(X) = p.

    Sample mean = \frac{x_1 + x_2 + \, .... + x_n}{n}.

    So use p = \hat{p} = \frac{x_1 + x_2 + \, .... + x_n}{n} as the estimator.

    1. b. Show whether or not E(\hat{p}) = p.

    -----------------------------------------------------------------------------------------

    2. a. E(X) = \alpha \, \beta \Rightarrow \alpha = \frac{E(X)}{\beta}.

    Sample mean = \frac{x_1 + x_2 + \, .... + x_n}{n}.

    So use \alpha = \hat{\alpha} = \frac{x_1 + x_2 + \, .... + x_n}{n \, \beta} as the estimator.

    2. b. Show whether or not E(\hat{\alpha}) = \alpha.
    Last edited by mr fantastic; January 10th 2009 at 11:44 PM. Reason: Added the questions from the attachment
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    Unbiased Estimation and Method of Moments 2

    Have you answered letter b. which is verifying if the moments estimator of p is unbiased for p. and the other one is in gamma distribution. Thanks!
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    Quote Originally Posted by aadbaluyot View Post
    Have you answered letter b. which is verifying if the moments estimator of p is unbiased for p. and the other one is in gamma distribution. Thanks!
    I have shown you how to answer letter b. in both questions and have given you the answer to part a., without which b. can't be done.

    If you show your working and say where you're stuck I will be able to give more help.
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    I have verified that the methods of moments estimators of p and α are unbiased for p and α. Am i right? My computation is attached here... Thanks for the big help.
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    Quote Originally Posted by aadbaluyot View Post
    I have verified that the methods of moments estimators of p and α are unbiased for p and α. Am i right? My computation is attached here... Thanks for the big help.
    Looks fine.
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  7. #7
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    for MR. Fantastic

    Sir, do you know any threads about bayesian estimation, maximum likelihood estimation and confidence interval that i can use for studying? Thanks!
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  8. #8
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    Quote Originally Posted by aadbaluyot View Post
    Sir, do you know any threads about bayesian estimation, maximum likelihood estimation and confidence interval that i can use for studying? Thanks!
    I suggest you search the MHF forums using key words.

    I also suggest you use Google.

    And a visit to the probability and statistics section of the library of the institute you study at would be time well spent.
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