2
$\begingroup$

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.

enter image description here

$\endgroup$
  • $\begingroup$ Please write your question as it is usual here in mathjax, see the help page for it. $\endgroup$ – Karl Dec 27 '14 at 13:17
  • $\begingroup$ Please write your question in Latex and then post. $\endgroup$ – Samrat Mukhopadhyay Dec 27 '14 at 13:17
  • $\begingroup$ 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. $\endgroup$ – Did Dec 28 '14 at 17:07
1
$\begingroup$

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 $x_1\gt1$, $\hat\theta_{\text{MLE}}(x_1)=0$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.