Finding the MLE if the second derivative is not less than 0

Question

Let $X_1,\,X_2,\,X_3,\ldots ,X_n$ represents a random sample from each of the distributions.

I have a certain function :

$$f(x;\theta)=\begin{cases}e^{\theta-x}& x\geq\theta \\ 0 & x<\theta\end{cases}$$

for $\theta\in(-\infty,\infty)$.

Finding the MLE!

My answer is the following :

$$\begin{align} L(\theta;\,x_i)&=e^{-\sum_{i=1}^n(x_i-\theta)}\\ \ln{L(\theta;\,x_i)}&=-\sum_{i=1}^n(x_i-\theta)\\ \ln{L(\theta;\,x_i)}&=n\theta-\sum_{i=1}^n x_i\\ D_\theta\ln{L(\theta;\,x_i)}&=n>0\\ D_{\theta\theta}\ln{L(\theta;\,x_i)}&=0 \end{align} $$

But i'm having trouble to find $\hat \theta$

My professor said that we must use ordered statistics when we meet this case. I mean, when the second derivative is zero or greater than zero.

But how to use ordered statistics for finding the estimator?

Answer 1

I think the second derivative is unnecessary.

Answer 2

Hint:

• The likelihood is zero if any of the $x_i$s is less than $\theta$

• The likelihood is a positive and increasing function of $\theta$ (you have shown it has a positive first derivative), provided that $\theta$ is less than or equal to all the $x_i$s

• So the likelihood is maximised when $\theta$ is ...