Show that the minimal sufficient statistics for the shifted exponential is complete for $n = 2$

Question

If we had $Y_i$, $i = 1, 2, ..., n$ are $iid$ and have the density $$f(y) = \lambda e^{-\lambda(y - \mu)} I_{y > \mu} , ~~~y >0$$ where $\lambda >0,~ \mu >0$ are unknown parameters.

I was able to show through factorization theorem that the minimum order statistic $T_1 = Y_{(1)}$ and $T_2 = \sum_i^n y_i$ are sufficient statistics. How would I go about showing that these are complete for $n = 2$?

Is the idea to find the distributions of $T_1$ and $T_2$, and then showing that $$E(g(T_1)) = 0 \implies g(T_1) = 0$$ and $$E(g(T_2)) = 0 \implies g(T_2) = 0?$$ It seems it should be simpler than this considering $n = 2$.

Answer 1

Since this is self-study question, I will not provide any solution but rather jot down the general approach to attack the problem.

It has been shown $(T_1, T_2):= \left(Y_{(1)}, \sum Y_i\right)\equiv \left(Y_{(1)}, \sum \left(Y_i-Y_{(1)}\right)\right)$ are jointly sufficient for $(\mu, \lambda).$

What should be the approach to show $(T_1, T_2)$ is jointly complete?

$\bullet$ Stick to the definition (unlike what OP attempted): Show that $$\mathbb E_{\mu,\lambda}[f(T_1, T_2)] = 0 \implies f\equiv 0 ~~\textrm{a.s.}\tag 1\label 1$$

$\bullet$ Show that $T_1, ~T_2$ are independently distributed. Find out their respective distributions.

$\bullet$ For fixed $\lambda,$ apply Fubini's theorem in $\eqref 1$ to get $$\int\underbrace{\left[\int f(t_1, t_2)f_{T_2}(t_2)~\mathrm dt_2\right]}_{=~\mathbb E_\lambda[f(t_1, T_2)]}f_{T_1}(t_1)~\mathrm dt_1 = 0\tag 2\label 2$$ for all $\mu.$

$\bullet$ Conclude $\mathbb E_\lambda f(t_1, T_2) = 0~~\textrm{a.s.}$ (How?)

$\bullet$ Finally show the above conclusion is true for all $\lambda.$ The rest follows. (How?)

$\rm NB.$ The approach is based on the assumption that $\mu\in \mathbb R, ~\lambda \in \mathbb R_{+}.$ Can OP modify (Is it necessary or not? Check.) the argument for $\mu\in\mathbb R_{+}?$

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Reference:

$\rm [I]$ Theory of Point Estimation, E. L. Lehmann, George Casella, Springer-Verlag, $1998,$ sec. $1.6,$ p. $43.$