 Solution

The Lehmann–Scheffé theorem states that an unbiased estimator which is a function of a complete sufficient statistic is the unique minimum-variance unbiased estimator; equivalently conditioning any unbiased estimator on such a statistic produces it.

Use the unbiased estimator $I=1_{X_1>x}$. Given $T_n=t$, the positive sample vector is uniform on the simplex ${\sum }_i{X}_i=t$: the joint exponential density is constant there, independently of $\lambda$. For $t<x$ the conditional probability is zero. For $t\ge x$, translating the first coordinate by $x$ changes the simplex size from $t$ to $t-x$; its $(n-1)$-dimensional volume ratio is $(1-x/t{)}^{n-1}$. Thusincluding $n=1$, when this is the indicator $1_{T_1\ge x}$ up to a null boundary. Conditional expectation preserves unbiasedness, and completeness from part (i) proves the displayed estimator is the unique minimum-variance unbiased estimator.

It does not attain the Cramér-Rao bound. Equality in that bound requires the centred unbiased estimator to be proportional to the score function, hence affine in $T_n$ at fixed $\lambda$. This estimator is zero on a positive-probability interval and positive elsewhere, so is not affine; the $n=1$ step function is not affine either.

For the last assertion, the remaining observations are independent of the first $k$. Conditional on $T_k$, the internal proportions of those first $k$ are independent of the remaining sum, soNow the tower property for the nested information $\sigma (T_n)\subseteq \sigma (T_k,T_n)$ givesConditioning first on both sums is essential; the sigma-fields generated by $T_k$ and $T_n$ alone are not nested.