  Peng (2008)(\cite{P08b}) proved the Central Limit Theorem under a sublinear
expectation:
  \textit{Let $(X_i)_{i\ge 1}$ be a sequence of i.i.d random variables under a
sublinear expectation $\hat{\mathbf{E}}$ with
$\hat{\mathbf{E}}[X_1]=\hat{\mathbf{E}}[-X_1]=0$ and
$\hat{\mathbf{E}}[|X_1|^3]<\infty$. Setting
$W_n:=\frac{X_1+\cdots+X_n}{\sqrt{n}}$, we have, for each bounded and Lipschitz
function $\varphi$,
\[\lim_{n\rightarrow\infty}\bigg|\hat{\mathbf{E}}[\varphi(W_n)]-\mathcal{N}_G(\varphi)\bigg|=0,\]
where $\mathcal{N}_G$ is the $G$-normal distribution with
$G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]$, $a\in \mathbb{R}$.}
  In this paper, we shall give an estimate of the rate of convergence of this
CLT by Stein's method under sublinear expectations:
  \textit{Under the same conditions as above, there exists $\alpha\in(0,1)$
depending on $\underline{\sigma}$ and $\overline{\sigma}$, and a positive
constant $C_{\alpha, G}$ depending on $\alpha, \underline{\sigma}$ and
$\overline{\sigma}$ such that
\[\sup_{|\varphi|_{Lip}\le1}\bigg|\hat{\mathbf{E}}[\varphi(W_n)]-\mathcal{N}_G(\varphi)\bigg|\leq
C_{\alpha,G}\frac{\hat{\mathbf{E}}[|X_1|^{2+\alpha}]}{n^{\frac{\alpha}{2}}},\]
where $\overline{\sigma}^2=\hat{\mathbf{E}}[X_1^2]$,
$\underline{\sigma}^2=-\hat{\mathbf{E}}[-X_1^2]>0$ and $\mathcal{N}_G$ is the
$G$-normal distribution with
\[G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]=\frac{1}{2}(\overline{\sigma}^2a^+-\underline{\sigma}^2a^-),
\ a\in \mathbb{R}.\]}
