find $\theta_{MLE}$ for a function

Question

For $$ f(x;\theta)=(\theta+1)x^{-\theta-2} $$ find the maxmimum likelihood estimators (MLEs) for $\theta$ based on a random sample of size $n$. My work so far: $$ \begin{align} \prod_{i=1}^n \log(f(x_i;\theta)) &= \sum \log(\theta+1)-\log(x_i)(\theta+2) \\ &=n\log(\theta+1)-(\theta+2)\sum \log(x_i) \end{align} $$ Now take the derivative, and set it equal to 0: $$ \begin{align} \frac{d}{dx} \log(f(x;\theta)) &= 0-(\theta+2)\sum \frac{1}{x_i} =0 \\ &=\sum\frac{1}{x_i}=0 \end{align} $$ I cant make any sense of this. What exactly have I done here? Did something go wrong? Shouldnt I get an equation with $\theta$ in it?

Answer 1

You should take the derivative with respect to $\theta$.

Also, $\prod_{i=1}^n \log(f(x_i;\theta))$ should be $\sum_{i=1}^n \log(f(x_i;\theta))$