# Statistics: parameter estimation

Let $X$ be a random variable characterized by the following density function:

$f(x; \theta) = ((\theta + x) / (\theta + 1)) * exp (-x)$, if $x >= 0$

$f(x; \theta) = 0$, if $x < 0$

Assuming that 0 <= theta <= 4, determine a maximum likelihood estimate of the parameter theta based on the realization sample x1 = 1/2.

This is my partial solution, but I cannot do the first derivative...

Thank you for considering my request.

• Please write your question as it is usual here in mathjax, see the help page for it.
– Karl
Commented Dec 27, 2014 at 13:17
• Please write your question in Latex and then post. Commented Dec 27, 2014 at 13:17
• How comes that your likelihood involves unspecified quantities $x_i$ for $1\leqslant i\leqslant n$ when the question specifies that the sample is $x_1=1/2$ (hence of size $1$? Please revise.
– Did
Commented Dec 28, 2014 at 17:07

For every $x_1\lt1$, the likelihood $L(\theta;x_1)=f(x_1;\theta)$ is an increasing function of $\theta$ hence, assuming that $\theta$ is rectricted to some interval $[0,\theta_*]$, one gets $\hat\theta_{\text{MLE}}(x_1)=\theta_*$. Likewise, for every $x_1\gt1$, $\hat\theta_{\text{MLE}}(x_1)=0$.