# MLE for lower bound of Uniform Distribution

Let $X_1$, $X_2$, . . . , $X_n$ be a random sample from a $Uniform(θ, 1)$ population, where $θ < 1$.

(a) Find the MLE $\widehat{\theta}$ of $θ$.

(b) Find constants c and d (possibly depending on n) such that, c+d$\widehat{\theta}$ is unbiased for $θ$.

The likelihood of $θ$ being greater than $X_1$ is zero, but what is the likelihood of it being lower $X_1$. Is it the $θ^{-n}$ (same as when we try to estimate the higher limit of the Uniform Distribution.

I understand that $θ$ should be the minimum of the sample and 1 should be the max of the sample (ideally), but how to derive it mathematically?

The likelihood of a sample $\mathbf x=(x_1,\ldots,x_n)$ in $(0,1)$ is $$L(\theta\mid\mathbf x)=\prod_{k=1}^n\frac{\mathbf 1_{[\theta,1]}(x_k)}{1-\theta}=\frac{\mathbf 1_{\theta\leqslant\min\mathbf x}}{(1-\theta)^n}.$$ Surely you can deduce which value $\hat\theta(\mathbf x)$ of $\theta$ maximizes this.