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!
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##?
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##.
The Method of Moment Estimator is a statistical technique used to estimate the parameters of a population based on the moments of the data. It is a non-linear method that uses the moments of the data to find the best fit for the model.
The Method of Moment Estimator works by equating the theoretical moments of a population to the sample moments of the data. This creates a system of equations that can be solved to find the estimated values of the parameters.
The Method of Moment Estimator is easy to use and does not require any assumptions about the underlying distribution of the data. It also provides consistent estimates and has good performance for large sample sizes.
The Method of Moment Estimator may produce biased estimates if the underlying distribution of the data is not well represented by the chosen model. It also requires a sufficient number of moments to be known and used in the estimation process.
The Method of Moment Estimator differs from other estimation methods in that it does not require knowledge of the entire distribution of the data. It also does not require iterative calculations like maximum likelihood estimation, making it computationally simpler.