What is the method of moment estimator for θ?

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Homework Help Overview

The discussion revolves around finding the method of moments estimator for the parameter θ in a probability density function defined as f_θ=2x/θ^2 for 0≤x≤θ. Participants are exploring the relationship between sample moments and the parameter θ, particularly in the context of a random sample from the given distribution.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to derive the method of moments estimator by relating sample means to expected values. Questions are raised about the expected value of the sample and how it connects to θ. Some participants are calculating expected values and variances, while others are questioning how to properly express the sample mean in relation to the parameter.

Discussion Status

The discussion is active, with participants sharing calculations and seeking clarification on the method of moments. Some have provided specific calculations for expected values, while others are exploring the implications of those calculations. There is no clear consensus yet, but multiple lines of reasoning are being explored.

Contextual Notes

Participants note the lack of examples from the professor and express uncertainty about the method of moments, indicating a need for further guidance on the topic. There is also mention of the distinction between sample statistics and random variables in the context of the problem.

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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!
 
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mathmathRW said:

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##?
 
Ray Vickson said:
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##.
 
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?
 
mathmathRW said:
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## ?
 

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