# Method of Moments (Beta Dist)

1. Nov 18, 2012

### mrkb80

1. The problem statement, all variables and given/known data
Let $X_1,...,X_n$ be iid with pdf $f(x;\theta) = \theta x^{\theta-1} , 0 \le x \le 1 , 0 < \theta < \infty$

Find an estimator for $\theta$ by method of moments

2. Relevant equations

3. The attempt at a solution
I know I need to align the first moment of the beta distribution with the first moment of the sample ($\bar{x}$ or $\dfrac{\Sigma_{i=1}^n x_i}{n}$)

The beta distribution has a first moment of $\dfrac{\alpha}{\alpha + \beta}$

I guess my problem is figuring out what my should be alpha and beta from the given pdf, from there it is simply just $\bar{x}=\dfrac{\alpha}{\alpha + \beta}$ and then solving for $\theta$ Any advice?

I did try to take the expected value of the pdf and set it equal to x bar, but I think that is not the correct answer ( I got something like $\hat{\theta}=\dfrac{\bar{x}}{1-\bar{x}}$ )

I also attempted this using MLE and got something like $\hat{\theta}=\dfrac{-n}{\Sigma_{i=1}^n \ln{x_i}}$ but I would also like to solve this problem with method of moments.

Last edited: Nov 18, 2012
2. Nov 18, 2012

### haruspex

If you represent it as a case of a Beta distribution, you get α=θ, β=1, yes? And that gives you the same result as you obtained directly. What makes you think it is wrong?

3. Nov 19, 2012