What is the method of moment estimator for θ?

[mathmathRW]

Homework Statement

Let $X_1, X_2, ..., X_n$ be a random sample from $f_θ=2x/θ^2$ , $0≤x≤θ$.
Find a maximum likelihood estimator for θ. Find the method of moment estimator for θ.

The Attempt at a Solution

I have already found that the MLE is max{$x_i$}. I just need to find the method of moments estimator. My professor hasn't given any examples on this and everything I have found online seems completely different. I would appreciate some guidance on this one!

 

[Ray Vickson]

Homework Statement

Let $X_1, X_2, ..., X_n$ be a random sample from $f_θ=2x/θ^2$ , $0≤x≤θ$.
Find a maximum likelihood estimator for θ. Find the method of moment estimator for θ.

The Attempt at a Solution

I have already found that the MLE is max{$x_i$}. I just need to find the method of moments estimator. My professor hasn't given any examples on this and everything I have found online seems completely different. I would appreciate some guidance on this one!

What is $EX_i$ in terms of $\theta$? What is $E \sum_{i=1}^n X_i / n$? So, what function of $\theta$ is estimated by the mean sample value $\sum_{i=1}^n x_i / n$?

 

[mathmathRW]

What is $EX_i$ in terms of $\theta$? What is $E \sum_{i=1}^n X_i / n$? So, what function of $\theta$ is estimated by the mean sample value $\sum_{i=1}^n x_i / n$?

I calculated $E(X)=2θ/3$ and $E(X^2)=θ^2/2$. I am not sure how to find $\sum_{i=1}^n x_i / n$.

 

[mathmathRW]

Ok, I have been looking online some more. Should I find $σ^2(X)$ ? It looks like maybe $∑X_i^2/n=σ^2+[E(X)]^2$. Is that the method of moments estimator?

I have found $σ^2(X)=θ^2/18$ and $∑X_i^2/n=σ^2+[E(X)]^2=θ^2/(2n)$.

Am I on the right track?

 

[Ray Vickson]

I calculated $E(X)=2θ/3$ and $E(X^2)=θ^2/2$. I am not sure how to find $\sum_{i=1}^n x_i / n$.

You find $\sum_{i=1}^n x_i / n$ by taking a sample of size n, measuring the resulting $x_i$ values and then computing the sum. On the other hand, the RANDOM VARIABLE $\sum_{i=1}^n X_i /n$ is a different animal completely. As a random variable, it has a certain mean and variance, etc. What are these values, expressed in terms of $n$ and $\theta$ ?