Likelihood Function for the Uniform Density $(\theta, \theta+1)$ Let the random variable X have a uniform density given by $f(x;\theta)\sim R(\theta,\theta+1)$.
What is the likelihood function according to the samples $X_1,\ldots,X_n$?
The question is much like Likelihood Function for the Uniform Density. But there seems no satisfying answer. So far I get $X_{\max}-1\le\theta\le X_{\min}$ and my guess is $\hat\theta=\frac{X_{\max}+X_{\min}-1}{2}$. Am I right? I need a theoretical explanation.

Here a two thoughts


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*If any $X_i$ lies outside of $[\theta,\theta+1]$, then the likelyhood function is zero

*If all the $X_i$ lie inside $[\theta_1,\theta_1+1]$ and also inside $[\theta_2,\theta_2+1]$, then the likelyhoods are the same, because $f(x,\theta_1) = f(x,\theta_2)$ for those $x$ which lie in the intersection of the two intervals

