Given $X_1, X_2, \ldots , X_n\sim \operatorname{i.i.d. DU}(K)$. Prove that the rate of convergence of $\widehat{K}=X_{(1)}+X_{(n)}-1$ is $\infty$.
I think I want to start by showing that $X_{(1)} \xrightarrow{P} 1$ so $\widehat{K}\xrightarrow{P}1+K-1=K$, but I don't know how to begin rigorously or how to show the rate of convergence is infinite.
The words "the rate of convergence is infinite" can be interpreted in different ways. For example, they can be treated like this: for any $\alpha>0$ $$ n^\alpha(\hat K-K) \xrightarrow{P} 0. $$ Use the definition of convergence in probability to show that for any $\alpha>0$ $$\tag{1}\label{1} n^\alpha(X_{(1)}-1) \xrightarrow{P} 0 \ \text{ and } \ n^\alpha(X_{(n)}-K) \xrightarrow{P} 0. $$ Say, for any $\varepsilon>0$ $$ \mathbb P\left(n^\alpha|X_{(1)}-1|\geq \varepsilon\right)=\mathbb P\left(X_{(1)}\geq 1+\frac{\varepsilon}{n^\alpha}\right)\leq \mathbb P\left(X_{(1)}\geq 2\right). $$ Prove that the last probability tends to zero, and you will get the first statement in (\ref{1}) proved. The same for the second statement.