MLE for a uniform distribution

Question

Let $X_1,\dots, X_n$ be a sample of independent random variables with uniform distribution $U(θ,θ+1)$, where $θ$ is positive. Then what is the MLE of $θ$?

The probability density function is $1$ and I don't know what to do.

Answer 1

The density function is not exactly $1$. Precisly, it is $f_X(x)=\mathbf{1}_{[\theta,\theta+1]}(x)$.

Then the likelihood function is given by: $$l(\theta;x_1,\dots,x_n)=\Pi_{i=1}^n\mathbf{1}_{[\theta,\theta+1]}(x_i)$$

This function is $1$ if $\min x_i\geq\theta$ and $\max x_i\leq\theta+1$, and $0$ otherwise. If you see this as a function of $\theta$, this function is $\mathbf{1}_{[\max x_i-1,\min x_i]}(\theta)$.

So the maximum of $l(\theta;x_1,\dots,x_n)$ is not unique, it can be any value between $\max x_i-1$ and $\min{x_i}$.

So, for example $\hat{\theta}=\min X_i$ is a MLE of $\theta$.