I have a likelihood function $$ L(\theta;\mathbf x) = \frac{\prod x_i^{\nu-1} \exp\left( -\sum x_i/\theta \right) }{\theta^{\nu n} [\Gamma(\nu)] } \qquad x>0 $$
It gets log-transformed into the following formula $$ \ln L(\theta;\mathbf x) = \text{constant} - \frac{n\overline x} \theta - \nu\theta\ln\theta $$
Two questions:
- I get the same result when I perform the transformation myself, except in addition to the above result I get an extra term $n\bar{x}(\nu-1)$ — why is it not supposed to be there?
- Also I get ${}-\text{const}$ rather than ${}+\text{const}$, but I suppose because it is an arbitrary constant value, then either $+$ or $-$ works?
In this context, "constant" means not depending on $\theta.$ All terms that do not depend on $\theta$ are constant. In particular, often the next thing one does after taking the logarithm is differentiating with respect to $\theta,$ and then every term not depending on $\theta$ vanishes.