Let $X_{1}, \ldots, X_{n}$ be a random sample from a probability density function $f(x \mid \theta)$, where $\theta$ is an unknown real-valued parameter which is assumed to have a prior density $\pi(\theta)$. Determine the optimal Bayes point estimate $a\left(X_{1}, \ldots, X_{n}\right)$ of $\theta$, in terms of the posterior distribution of $\theta$ given $X_{1}, \ldots, X_{n}$, when the loss function is

$$L(\theta, a)= \begin{cases}\gamma(\theta-a) & \text { when } \theta \geqslant a \\ \delta(a-\theta) & \text { when } \theta \leqslant a\end{cases}$$

where $\gamma$ and $\delta$ are given positive constants.

Calculate the estimate explicitly in the case when $f(x \mid \theta)$ is the density of the uniform distribution on $(0, \theta)$ and $\pi(\theta)=e^{-\theta} \theta^{n} / n !, \theta>0$.