Define
$$
R_X(\mu)=\sup_{\|v\|_2\leq1}
|\operatorname{MOM}_k(Xv)-\mu^Tv|.
$$
For every unit-ball vector $v$, the <triangle inequality> gives
$$
|v^T(\widehat\mu-\mu_0)|
\leq|\operatorname{MOM}_k(Xv)-\widehat\mu^Tv|
+|\operatorname{MOM}_k(Xv)-\mu_0^Tv|.
$$
Taking the supremum and using optimality, $R_X(\widehat\mu)\leq R_X(\mu_0)$, yields
$$
\|\widehat\mu-\mu_0\|_2
\leq R_X(\widehat\mu)+R_X(\mu_0)
\leq2R_X(\mu_0).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 223 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-223/4/b/solution