Suppose that $Y_1, \dots, Y_n$ are i.i.d observations from the density $f(y, \theta, \beta) = \beta e^{-\beta(y - \theta)}I_{[y \geq\theta]}$
where $\beta \gt 0$, $\theta \in \mathbb{R}$ are unknown parameters.
Let $(T_1, T_2) = (\min(Y_1, \dots, Y_n), \bar{Y})$. I want to show that $T_2 - T_1$ is independent of $T_1$ for all values of $(\theta, \beta)$. I already proved that $T_1$ is complete sufficient for $\theta$ when $\beta$ is fixed and known, and that $T_2$ is complete sufficient for $\beta$ when $\theta$ is fixed and known. Clearly, $T_2 - T_1$ is an ancillary statistic of $\theta$.
Since $T_1$ is complete sufficient for $\theta$ when $\beta$ is fixed and known, can we just use Basu's theorem to conclude that $T_2 - T_1$ is independent of $T_1$ for all values of $(\theta, \beta)$?