MLE problem - the likelihood function has no maximum.

Question

The probability density function is:

$f(x)=e^{\theta -x}, \ 0 \le \theta \le x $

Given an n-element sample, the likelihood function is:

$$L(\theta)=\exp \left( n\theta - \sum_{i=1}^n x_i \right)$$

Since the function has no maximum, but increases with $\theta$, I would take the estimator to be $\hat{\theta}=\max X_i$, following an example I did in class. But the answers I have (not 100% correct) say to take the minimum. Can someone shed some light on this?

Answer 1

This will be clear if you write the likelihood function more carefully as \begin{align} L(\theta) &= e^{ \left( n\theta - \sum_{i=1}^n x_i \right) }\prod_{i=1}^n 1_{\{ \theta \le x_i\}} \\ &= e^{ \left( n\theta - \sum_{i=1}^n x_i \right) }1_{\{ \theta \le \min\limits_{1\le i\le n} x_i\}} \end{align}

As you correctly noted, $L(\theta)$ is an increasing function of $\theta$ on the interval $[0, \min x_i]$. And for $\theta > \min x_i$, the likelihood is zero.

Thus, $\hat{\theta} = \min X_i$.