Let $\theta > 0$ be some model parameter for which properties (bias, ...) of an estimate are studied via simulations.

For a given data set, an estimate $\hat{\theta}_i$ of $\theta$ can be obtained by maximising a likelihood function.

As $\theta$ is positive, however, I perform the maximisation over $\eta = \log(\theta) \in R$ and I set $\hat{\theta} = \exp(\hat{\eta})$.

After running $N$ simulations, how should I compute the mean value,

• $\bar{\hat{\theta}} = \frac{1}{N} \sum_{i=1}^N \hat{\theta}_i$, or
• $\bar{\hat{\theta}} = \frac{1}{N} \sum_{i=1}^N \exp(\hat{\eta}_i)$?