Good evening everyone. I might need some help on something. Suppose we have $n$ independent variables from $U[0,1+θ]$ and suppose also that $λ=1+θ$. The estimator of $θ$ is $\bar{θ}=2 \bar{X} - 1$. Ι need to find the bias of $\bar{θ}$ knowing that the distribution of $\bar{X}$ of $n$ variables, which they are from $U[0,1]$ , is the Bates distribution, which has the following probability density function: $$f(x)=\frac{n}{2(n-1)!} \sum_{k=0}^n (-1)^k \binom{n}{k}(nx - k)^{(n-1)} sgn(nx -k)$$
where $\begin{cases}sgn(nx-k)&=&-1,\ & \text{if} \ \ nx<k\\ sgn(nx-k)&=& \ \ \ 0, & \text{if} \ \ nx=k\\ sgn(nx -k)&=& \ \ \ 1, &\text{if} \ \ nx > k \ \end{cases}$.