Suppose $\{X_{i}\}_{i=1}^n\overset{i.i.d}{\sim}X$, where $X$ has density $$f_{X}(x)=\frac{1}{b}\exp\left\{\frac{x-a}{b}\right\},x>a$$ What is the UMVUE of $\mathbb{P}(X_1<u)$?
Here is what I've attempted.
Denote $S=\sum_{i=1}^n(X_i-X_{(1)})$. We know that $T=(X_{(1)},S)$ is jointly sufficient + complete statistic for $(a,b)$. Thus $\mathbb{P}(X_1<u|X_{(1),S})$ is the UMVUE of $(a,b)$.
Next is to calculate $\mathbb{P}(X_1<u|X_{(1),S})$. I have $$\mathbb{P}(\frac{X_1-X_{(1)}}{S}<\frac{u-X_{(1)}}{S}|X_{(1),S})\overset{Basu}{=}\mathbb{P}(\frac{X_1-X_{(1)}}{S}<\frac{u-x_{(1)}}{s}),$$ where we used the fact that $\frac{X_1-X_{(1)}}{S}$ is ancillary to $T$, which implies independence between them by Basu's theorem.
However, I got stuck on getting the distribution of $Z=\frac{X_1-X_{(1)}}{S},$ thus I cannot calculate the probability.
