Maximum Likelihood Estimator - What did I do wrong?

For a X_1,…,X_n random sample from a uniform distribution from [0, 2*theta+1], I was asked to find the MLE. The pdf: f(x|theta) = 1/(2*theta+1) for x in [0, 2*theta+1] and 0 otherwise.

I determined the likelihood(theta)=L(theta)=(2*theta+1)^-n
Then the log-likelihood to be -n*(2*theta+1), finding the first derivative and setting to 0 to maximize is ineffective. Examining the likelihood function, X_(n), the MAX(X_1,…,X_n) should maximize the likelihood function.

Professor says I am wrong in developing my likelihood function. Can anybody help?

Thanks in advance.


Ambassador to the humans
Your likelihood is (almost) correct but it's missing the key piece for this problem. Is it possible for (2*theta+1) to be smaller than any of the observed data values?


Ambassador to the humans
Then your likelihood needs to be 0 if 2*theta+1 is less than than any of the values (this can be simplified to a single comparison). You might consider using an indicator function.