Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 29K / e / Solution

No. Every estimator in parts (b) and (c) has $a\ne 1$, so its risk contains the nonconstant quadratic term
$((a-1)\theta +b{)}^2$ and has infinite supremum over $\theta \in \mathbb{R}$. The estimator ${\overline{X}}_n$ has constant risk $1/n$, so those estimators cannot be minimax.