Edit: i see this post similar question but I don't see how to connect it to the cumulative distribution function (CDF).
I'm working on a problem involving the derivation of the Maximum Likelihood Estimator (MLE) for a uniform distribution, and I'm a bit stuck. The task also involves calculating the cumulative distribution function (CDF) and the probability density function (PDF) for a particular scenario. Here's what I'm trying to figure out:
Consider $( X_1, ..., X_n )$ ~$Uniform(0, \theta)$ from a uniform distribution spanning the interval $[0, \theta ]$ , where $ \ ( \theta > 0 )$ is an unknown parameter.
The goal is to show that the maximum likelihood estimator (MLE) for parameter $ \theta $ is ( $ \hat{\theta} = \max\{X_1, ..., X_n\} )$
Furthermore, I need to calculate the CDF and PDF for the estimator $\hat{\theta}$, assuming $\theta > 0 $ is a real random parameter.
I understand the basics of MLE but applying it to the uniform distribution, especially in the context of extracting the maximum value from a set of samples, has proven to be a challenge. Transitioning from the concept of MLE to deriving the distribution functions for $\hat{\theta}$ is where I'm particularly stuck.
Could anyone provide guidance on how to approach deriving the MLE for $\theta $ and how to calculate the CDF and the PDF for $\hat{\theta}$ in this case?
Any step-by-step explanations, examples, or resources that you could share would be immensely helpful.
Thank you for your time and assistance!