Hi please help me with this problem. With the random samples $X_1,\dots,X_n$ from $\operatorname{Exp}(\mu, \sigma)$, I need to attain the MVUE of $\eta = \mathbb{P}(X_1>a)$. I used the Lehmann-Scheffe and Basu theorem to get close to my goal. However, I can't attain the distribution of ancillary statistic. My solution so far is as below.
We have CSS of $(\mu, \sigma)$, $T = (X_{(1)}, \sum^n_{i=1}(X_i - X_{(1)})$ and since $\eta = \mathbb{E}_\theta\mathbb{I}(X_1>a)$} by Lehmann-Scheffe, $$\eta^{MVUE} = \mathbb{P}(X_1>a|X_{(1)}=x, S=s) \text{ where $S=\sum^n_{i=1}(X_i - X_{(1)})$}\\ =\mathbb{P}\Big(\frac{X_1-X_{(1)}}{S}>\frac{a-x}{s}\Big)\qquad (\because \text{Basu})\\ =\begin{cases} 1 \ (a < x) \\ \mathbb{P}(\frac{X_1-X_{(1)}}{S}>\frac{a-x}{s}) \ (a\geq x) \end{cases}\\ =\begin{cases} 1 \ (a < x) \\ \mathbb{P}(\frac{X_1-X_{(1)}}{S}>\frac{a-x}{s}, X_1 > X_{(1)}) \ (a\geq x) \end{cases}\\ $$ Let us consider the distribution of the ancillary statistic from now.
WLOG take $\mu=0, \sigma=1$, and since $X_1 > X_{(1)}$, $X_{(1)} = \min_{2\leq i \leq n}X_i = \min_{1\leq i \leq m}Z_i$, where $Z_1,\dots,Z_m \sim \text{i.i.d. } \operatorname{Exp}(1)$.
Observe that $X_1, Z_{(1)}, \sum^m_{i=1}(Z_i-Z_{(i)})$ are independent by exponential spacing.
Then we have $$ \frac{X_1-X_{(1)}}{S}=\frac{X_1 - Z_{(1)}}{X_1 - Z_{(1)} + \sum^m_{i=1}(Z_i-Z_{(1)})} $$ As you can see by implementing this new $Z$ variable, the joint distribution we had which was dependent is now jointly independent. With the transformation of variables we also have $$ \operatorname{pdf}_{X_1-Z_{(1)}}(y) = \frac{m}{m+1}e^{-y}\mathbb{I}(y>0),\sum^m_{i=1}(Z_i-Z_{(1)}) \sim \Gamma(m-1,1) $$ My gut is telling me that this is Beta Distribution. However, I can't figure out its parameters. Can anyone help me through this? Also, if anyone can think of some easier way please let me know.
@StubbornAtoms answered the same question but I want to attain the "specific" beta distribution. But his solution explains why I split the variable of $X$'s into $X$'s and $Z$'s.
