Determine if an Estimator is Biased (Unusual Expectation Expression)

Define, $$ Y = -\ln X, $$ thus $$ F_Y(y) = \mathbb{P}(Y\le y)=\mathbb{P}(X\ge e^{-y}) = 1-F_X(e^{-y}), $$ hence, $$ f_X(x) = f_X(e^{-y})e^{-y}=\theta e^{-y (\theta - 1)}e^{-y} = \theta e^{-y\theta}, \quad y>0. $$ Therefore, $$ - \sum_{i=1}^n\ln X_i \sim \mathcal{G}amma(n,\theta). $$ As such, $$ \frac{n}{-\sum_{i=1}^n \ln X_i} \sim n\times \mathcal{I}nv\mathcal{G}amma(n, \theta), $$ then the expectation is $$ \mathbb{E}[\hat{\theta}_n] = \frac{n}{n-1}\theta > \theta, $$ namely, $$ \tilde{\theta}_n = \frac{n-1}{n}\hat{\theta}_n, $$ is an unbiased estimator of $\theta$.