Finding the Method of moments estimator? Having trouble finding E(Y^2)

In summary, the conversation discusses finding the method of moments estimator (MME) for theta in a random sample from a beta distribution with a specific probability density function. The individual mentions that they are having trouble finding the expected value of Y squared and asks for suggestions.
  • #1
laura_a
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0

Homework Statement


Let Y1, Y2, ... Yn be a random sample from the distribution with pdf
[tex] \frac{\Gamma(2 \theta)}{[\Gamma(\theta)]^2} (y^{\theta -1)(1-y)^{\theta -1}[/tex]
for [tex] 0 \leq y \leq 1 [/tex]

I have to find the MME for theta


Homework Equations



This is a beta distribution where m = n = [tex]\theta[/tex]


The Attempt at a Solution



Now I believe that E(Y) = [tex] \frac{m}{m+n} [/tex]

So I worked out that E(X) = 1/2 which means it doesn't depend on theta.

SO I need to find [tex] E(Y^2) [/tex] which I already know is
[tex] \frac {\theta + 1}{2(2 \theta +1)} [/tex]

but I just don't know how to get it. I must be missing a formula because if I just do E(Y^2) from what I have, I end up with

[tex] \frac{1}{4} [/tex]

I can't even begin to find the MME because I can't find [tex] E(Y^2) [/tex]

Can anyone suggest a path I should go down? Thanks :)
 
Last edited:
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  • #2
dont worry I've got it now!
 

What is the Method of Moments estimator?

The Method of Moments estimator is a statistical technique used to estimate the parameters of a probability distribution by equating the population moments to the sample moments. It is an alternative to the Maximum Likelihood Estimator and is often used when the likelihood function is difficult to derive or solve.

How is the Method of Moments estimator calculated?

The Method of Moments estimator is calculated by equating the first k population moments to the first k sample moments, where k is the number of parameters to be estimated. The resulting equations can then be solved to obtain the estimates for the parameters.

What are the advantages of using the Method of Moments estimator?

One advantage of using the Method of Moments estimator is that it is easy to understand and implement. It also does not require knowledge of the underlying distribution of the data, making it more robust in situations where the distribution is unknown or difficult to determine.

What are the limitations of the Method of Moments estimator?

One limitation of the Method of Moments estimator is that it may not always produce the most efficient estimates. It also assumes that the moments of the population and sample are equal, which may not always be the case. Additionally, it may not work well for small sample sizes or when the data is highly skewed.

How can I find E(Y^2) using the Method of Moments estimator?

To find E(Y^2) using the Method of Moments estimator, you would set the second population moment (E(Y^2)) equal to the second sample moment (1/n * ∑(Y_i^2)) and solve for the parameter of interest. However, this may not always be possible or may result in a complex solution, in which case alternative methods such as Maximum Likelihood Estimation may be more appropriate.

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