# Math Help - Unbiased Estimation and Method of Moments

1. ## Unbiased Estimation and Method of Moments

I need your help...Please read the document that I attached here...thanks!

2. 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 $\alpha$.

Originally Posted by aadbaluyot
I need your help...Please read the document that I attached here...thanks!

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 $E(\hat{p}) = p$.

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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 $E(\hat{\alpha}) = \alpha$.

3. ## 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!

4. Originally Posted by aadbaluyot
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.

5. 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.

6. Originally Posted by aadbaluyot
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.

7. ## 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!

8. Originally Posted by aadbaluyot
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.