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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!

maya cohen
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    Can you try to write down the joint density $f(x,\theta)$? As a function of $\theta$ and fix the observed value $x_1,\cdots,x_n$, on which interval it increases, decreases or remains a constant? You may check Casella&Berger's statistical inference section 5.4 for the pdf of the maximum(and more generally, order statistics). – GC40 Mar 16 '24 at 18:38
  • i see this post similar question but I don't see how to connect it to the cumulative distribution function (CDF). – maya cohen Mar 16 '24 at 18:50
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    If you know pdf, you know cdf by integration. – GC40 Mar 16 '24 at 18:52
  • sorry , yes my English is vary bad this is why I feel so difficult in this area – maya cohen Mar 16 '24 at 19:06

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