if $f(x) = a exp(-a (x-b))$ . Find sufficient statistic for b

Question

Intuitively I feel that the answer should be $\min(X_1,X_2,\ldots,X_n)$ where $X_i$'s are iid, but I don't know how to prove it.

Answer 1

The domain for this problem is very important. You haven't specified it but I'm assuming the problem is: find the sufficient statistic for $$f(x) = a\, exp[-a(x-b)] \, \textbf{1}\{x > b\}.$$

Then the joint pdf is $$f(x) = a^n exp\left[-a \sum_{i=1}^{n} x_{i}\right] \,exp[a\,b\,n] \, \prod_{i=1}^{n}\textbf{1}\{x_{i}>b\}.$$ Notice all the $x_{i}$ will be greater than $b$ as long as the minimum is greater than $b$, therefore the likelihood function is $$L(b) =a^n exp\left[-a \sum_{i=1}^{n} X_{i}\right] \,exp[a\,b\,n] \, \textbf{1}\{X_{1,n}>b\},$$ where $X_{1,n} = min(X_{1},...,X_{n})$. Then $X_{1,n}$ is a sufficient statistic. It is also the MLE.