Unbiased Estimation and Method of Moments

**aadbaluyot** · January 10th 2009, 07:34 PM

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

**mr fantastic** · January 10th 2009, 09:12 PM

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!

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

**aadbaluyot** · January 11th 2009, 12:04 AM

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!

**mr fantastic** · January 11th 2009, 12:09 AM

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.

**aadbaluyot** · January 11th 2009, 12:38 AM

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.

**mr fantastic** · January 11th 2009, 12:58 AM

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.

**aadbaluyot** · January 11th 2009, 04:42 PM

Sir, do you know any threads about bayesian estimation, maximum likelihood estimation and confidence interval that i can use for studying? Thanks!

**mr fantastic** · January 11th 2009, 05:29 PM

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.

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