Suppose we have doubly type-2 censored sample $X_a \le X_{a+1}\le \cdots\le X_b$ from exponential population $\exp(\mu,\sigma)$ then $(X_a, V)$ is the complete and sufficient statistics for $(\mu,\sigma)$, where $V=-(n-a)X_a+X_{a+1}+\cdots+(n-b+1)X_b=(n-a)(X_{a+1}-X_{a})+(n-a-1)(X_{a+2}-X_{a+1})+\cdots+(n-b+1)(X_b-X_{b-1})$.
From likelihood function $L$ of given sample, I am able to compute the distribution $f_{X_a}$ of $X_a$, where
$f_{X_a}(x)=\frac{n!}{(n-a)(a-1)!}{\left(1-e^{-\frac{x-\mu}{\sigma}} \right)}^{a-1}\frac{1}{\sigma}{\left(e^{-\frac{x-\mu}{\sigma}} \right)}^{n-a+1}$ $x>\mu$,
and I know from article-"Confidence intervals for the scale parameter of exponential distribution based on Type II doubly censored samples" page number $2046$, that $V$ follows $\Gamma(b-a,1/\sigma)$ gamma distribution with shape parameter $b-a$ and rate parameter $1/\sigma$.
Can someone provide some hint or some reference book about finding the distribution of $V$ for given doubly type-2 censored data?
\cdots. – joriki Jan 22 '24 at 07:36