SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \sup  _ {G _ {*} \in \Xi} \mathbb {E} _ {p _ {G _ {*}}} \left(\left(\lambda^ {*}\right) ^ {2} \| \left(\Delta \mu^ {*}, \Delta \Sigma^ {*}\right) \| ^ {2} \| \left(\widehat {\mu} _ {n}, \widehat {\Sigma} _ {n}\right) - \left(\mu^ {*}, \Sigma^ {*}\right) \| ^ {2}\right) \lesssim \frac {\log^ {2} n}{n}, \\ \sup  _ {G * \in \Xi} \mathbb {E} _ {p _ {G *}} \left(\| (\Delta \widehat {\mu} _ {n}, \Delta \widehat {\Sigma} _ {n}) \| ^ {2} \| (\Delta \mu^ {*}, \Delta \Sigma^ {*}) \| ^ {2} | \widehat {\lambda} _ {n} - \lambda^ {*} | ^ {2}\right) \lesssim \frac {\log^ {2} n}{n}. \tag {34} \\ \end{array}
$$
It is clear that the second result in (34) does not match with the second result in the conclusion of part (b) of the theorem. To circumvent this issue, we utilize the fact that  $G_{*} \in \Xi_{1}(l_{n})$ . Indeed, notice that  $(\widehat{\mu}_n, \widehat{\Sigma}_n) - (\mu^*, \Sigma^*) = (\Delta \widehat{\mu}_n, \Delta \widehat{\Sigma}_n) - (\Delta \mu^*, \Delta \Sigma^*)$ , we have
$$
\sup  _ {G _ {*} \in \Xi} \frac {\mathbb {E} _ {p _ {G _ {*}}} \left\|\left(\Delta \widehat {\mu} _ {n} , \Delta \widehat {\Sigma} _ {n}\right) - \left(\Delta \mu^ {*} , \Delta \Sigma^ {*}\right)\right\| ^ {2}}{\| \left(\Delta \mu^ {*} , \Delta \Sigma^ {*}\right) \| ^ {2}} \lesssim \frac {\log^ {2} n}{n \left(\lambda^ {*}\right) ^ {2} \| \left(\Delta \mu^ {*} , \Delta \Sigma^ {*}\right) \| ^ {4}} \rightarrow 0. \tag {35}
$$
Hence, by the AM-GM inequality, we have
$$
\begin{array}{l} \mathbb {E} _ {p _ {G _ {*}}} \| (\Delta \widehat {\mu} _ {n}, \Delta \widehat {\Sigma} _ {n}) \| ^ {2} \big (\widehat {\lambda} _ {n} - \lambda^ {*} \big) ^ {2} \\ \geq \frac {1}{2} \| (\Delta \mu^ {*}, \Delta \Sigma^ {*}) \| ^ {2} E _ {p _ {G _ {*}}} (\widehat {\lambda} _ {n} - \lambda^ {*}) ^ {2} - E _ {p _ {G _ {*}}} \| (\Delta \widehat {\mu} _ {n}, \Delta \widehat {\Sigma} _ {n}) - (\Delta \mu^ {*}, \Delta \Sigma^ {*}) \| ^ {2} (\widehat {\lambda} _ {n} - \lambda^ {*}) ^ {2} \\ = \frac {1}{2} \| (\Delta \mu^ {*}, \Delta \Sigma^ {*}) \| ^ {2} \left(E _ {p _ {G _ {*}}} (\widehat {\lambda} _ {n} - \lambda^ {*}) ^ {2} - \frac {E _ {p _ {G *}} \left\| (\Delta \widehat {\mu} _ {n} , \Delta \widehat {\Sigma} _ {n}) - (\Delta \mu^ {*} , \Delta \Sigma^ {*}) \right\| ^ {2} (\widehat {\lambda} _ {n} - \lambda^ {*}) ^ {2}}{\| (\Delta \mu^ {*} , \Delta \Sigma^ {*}) \| ^ {2}}\right) \\ \gtrsim \left\| \left(\Delta \mu^ {*}, \Delta \Sigma^ {*}\right) \right\| E _ {p _ {G _ {*}}} \left(\widehat {\lambda} _ {n} - \lambda^ {*}\right) ^ {2}, \tag {36} \\ \end{array}
$$
uniformly in  $G_*$ , where in the last inequality we use (35) combining with the fact that  $|\widehat{\lambda}_n - \lambda^*|$  is uniformly bounded by 2. Hence,
$$
\mathbb {E} _ {p _ {G *}} \left(\| (\Delta \mu^ {*}, \Delta \Sigma^ {*}) \| ^ {4} | \widehat {\lambda} _ {n} - \lambda^ {*} | ^ {2}\right) \lesssim \mathbb {E} _ {p _ {G *}} \left(\| (\Delta \widehat {\mu} _ {n}, \Delta \widehat {\Sigma} _ {n}) \| ^ {2} \| (\Delta \mu^ {*}, \Delta \Sigma^ {*}) \| ^ {2} | \widehat {\lambda} _ {n} - \lambda^ {*} | ^ {2}\right) \lesssim \frac {\log^ {2} (n)}{n},
$$
which is the conclusion of the theorem.
D.3 Proof of Theorem A.2
$$
d _ {5} \left(G _ {1}, G _ {2}\right) = \lambda_ {1} \| \left(\mu_ {1}, \Sigma_ {1}\right) - \left(\mu_ {2}, \Sigma_ {2}\right) \| ^ {4},
$$
$$
{d _ {6} (G _ {1}, G _ {2})} = {| \lambda_ {1} - \lambda_ {2} | | | (\Delta \mu_ {1}, \Delta \Sigma_ {1}) | | ^ {4}.}
$$
for any  $G_{1} = G_{1}(\lambda_{1},\mu_{1},\Sigma_{1})$  and  $G_{2} = G_{2}(\lambda_{2},\mu_{2},\Sigma_{2})$ . It is clear that  $d_6(G_1,G_2)$  satisfies weak triangle inequality while  $d_5(G_1,G_2)$  no longer satisfies weak triangle inequality. In particular, we have
$$
d _ {5} (G _ {1}, G _ {3}) + d _ {5} (G _ {2}, G _ {3}) \geq \frac {\min  \left\{d _ {5} (G _ {1} , G _ {2}) , d _ {5} (G _ {2} , G _ {1}) \right\}}{8}.
$$
A close investigation of Lemma 6.1 of [16] reveals that modified Le Cam method still works under this setting of  $d_5$  metric. More specifically, for any  $\epsilon >0$  the following holds
$$
\inf  _ {\widehat {G} _ {n} \in \Xi} \sup  _ {G \in \Xi_ {2} (l _ {n})} \mathbb {E} _ {p _ {G}} \left(d _ {5} ^ {2} (G, \widehat {G} _ {n})\right) \geq \frac {\epsilon^ {2}}{1 2 8} \left\{1 - V \left(p _ {G _ {1}} ^ {n}, p _ {G _ {2}} ^ {n}\right) \right\}
$$
where  $G_{1}, G_{2} \in \Xi_{2}(l_{n})$  such that  $d_{5}(G_{1}, G_{2}) \wedge d_{5}(G_{1}, G_{2}) \geq \epsilon / 4$ . From here, to achieve the conclusion of part (a), it suffices to demonstrate for any  $r < 1$  that
(i) There exists two sequences  $G_{1,n} = (\lambda_n, \mu_{1,n}, \Sigma_{1,n}) \in \Xi_2(l_n)$  and  $G_{2,n} = (\lambda_n, \mu_{2,n}, \Sigma_{2,n}) \in \Xi_1(l_n)$  such that  $d_5(G_{1,n}, G_{2,n}) \to 0$  and  $h(p_{G_{1,n}}, p_{G_{2,n}}) / d_5^r(G_{1,n}, G_{2,n})$  as  $n \to \infty$ .
(ii) There exists two sequences  $G_{1,n}^{\prime} = (\lambda_{1,n}, \mu_n, \Sigma_n) \in \Xi_2(l_n)$  and  $G_{2,n}^{\prime} = (\lambda_{2,n}, \mu_n, \Sigma_n) \in \Xi_1(l_n)$  such that  $d_6(G_{1,n}, G_{2,n}) \to 0$  and  $h(p_{G_{1,n}^{\prime}}, p_{G_{2,n}^{\prime}}) / d_6^r(G_{1,n}, G_{2,n})$  as  $n \to \infty$ .
Following the proof argument of Proposition D.1, we can quickly verify the above results. As a consequence, we reach the conclusion of part (a) of the theorem.
(b) From the discussion after Theorem 3.5, we can show that:
$$
\mathcal {Q} (G, G _ {*}) \asymp | \lambda - \lambda^ {*} | (\| \Delta \mu \| ^ {2} \| \Delta \Sigma \|) (\| \Delta \mu^ {*} \| ^ {2} \| \Delta \Sigma^ {*} \|) + (\| \mu - \mu^ {*} \| ^ {2} + \| \Sigma - \Sigma^ {*} \|) \| (\lambda (\| \Delta \mu \| ^ {2} + \| \Delta \Sigma \|) + \lambda^ {*} (\| \Delta \mu^ {*} \| ^ {2} + \| \Delta \Sigma^ {*} \|)).
$$
Hence, from Theorem 3.6 combining with Theorem 2.3, we have
$$
\begin{array}{l} \sup _ {G _ {*}} \mathbb {E} _ {p _ {G _ {*}}} (\lambda^ {*}) ^ {2} (\| \widehat {\mu} _ {n} - \mu^ {*} \| ^ {4} + \| \widehat {\Sigma} _ {n} - \Sigma^ {*} \| ^ {2}) (\| \Delta \mu^ {*} \| ^ {4} + \| \Delta \Sigma^ {*} \| ^ {2}) \lesssim \frac {\log^ {2} (n)}{n} \\ \sup _ {G _ {*}} \mathbb {E} _ {p _ {G _ {*}}} | \widehat {\lambda} _ {n} - \lambda^ {*} | ^ {2} (\| \Delta \widehat {\mu} _ {n} \| ^ {4} \| \Delta \widehat {\Sigma} _ {n} \| ^ {2}) (\| \Delta \mu^ {*} \| ^ {4} \| \Delta \Sigma^ {*} \| ^ {2}) \lesssim \frac {\log^ {2} (n)}{n}. \\ \end{array}
$$
Similar to the proof of Theorem A.1 and with the definition of  $\Xi_2(l_2)$ , we have
$$
\mathbb {E} _ {p _ {G _ {*}}} | \widehat {\lambda} _ {n} - \lambda^ {*} | ^ {2} (\| \Delta \widehat {\mu} _ {n} \| ^ {4} \| \Delta \widehat {\Sigma} _ {n} \| ^ {2}) \gtrsim (\| \Delta \mu^ {*} \| ^ {4} \| \Delta \Sigma^ {*} \| ^ {2}) \mathbb {E} _ {p _ {G _ {*}}} | \widehat {\lambda} _ {n} - \lambda^ {*} | ^ {2}
$$
uniformly in  $G_{*}\in \Xi_{2}(l_{2})$  . Hence,
$$
\sup _ {G _ {*} \in \Xi_ {2} (l _ {2})} \mathbb {E} _ {p _ {G _ {*}}} | \widehat {\lambda} _ {n} - \lambda^ {*} | ^ {2} (\| \Delta \mu^ {*} \| ^ {8} \| \Delta \Sigma^ {*} \| ^ {4}) \lesssim \frac {\log^ {2} (n)}{n}.
$$
E Proofs for Auxiliary Results
Lemma E.1. For any  $r \geq 1$ , we define
$$
\begin{array}{l} D _ {r} (G, G _ {*}) = \lambda \| (\Delta \mu , \Delta \Sigma) \| ^ {r} + \lambda^ {*} \| (\Delta \mu^ {*}, \Delta \Sigma^ {*}) \| ^ {r} \\ - \quad \min  \left\{\lambda , \lambda^ {*} \right\} \left(\| (\Delta \mu , \Delta \Sigma) \| ^ {r} + \| (\Delta \mu^ {*}, \Delta \Sigma^ {*}) \| ^ {r} - \| (\mu , \Sigma) - (\mu^ {*}, \Sigma^ {*}) \| ^ {r}\right) \\ \end{array}
$$
for any  $G$  and  $G_*$ . Then, we have  $W_r^r(G, G_*) \asymp D_r(G, G_*)$  for any  $r \geq 1$  where  $W_r$  is the  $r$ -th order Wasserstein distance.
Proof. Without loss of generality, we assume throughout the lemma that  $\lambda < \lambda^{*}$ . Therefore, we obtain from the formulation of  $D_{r}(G,G_{*})$  that
$$
D _ {r} (G, G _ {*}) = (\lambda^ {*} - \lambda) | | (\Delta \mu^ {*}, \Delta \Sigma^ {*}) | | ^ {r} + \lambda | | (\mu , \Sigma) - (\mu^ {*}, \Sigma^ {*}) | | ^ {r}.
$$
Direct computation of  $W_r^r (G,G_*)$  yields three distinct cases:
Case 1: If  $||(\Delta \mu, \Delta \Sigma)||^r + ||(\Delta \mu^*, \Delta \Sigma^*)||^r \geq ||(\mu, \Sigma) - (\mu^*, \Sigma^*)||^r$ , then
$$
\begin{array}{l} W _ {r} ^ {r} (G, G _ {*}) = \lambda | | (\Delta \mu , \Delta \Sigma) | | ^ {r} + \lambda^ {*} | | (\Delta \mu^ {*}, \Delta \Sigma^ {*}) | | ^ {r} \\ - \min  \left\{\lambda , \lambda^ {*} \right\} \left(\left| \left| (\Delta \mu , \Delta \Sigma) \right| \right| ^ {r} + \left| \left| (\Delta \mu^ {*}, \Delta \Sigma^ {*}) \right| \right| ^ {r} - \left| \left| (\mu , \Sigma) - (\mu^ {*}, \Sigma^ {*}) \right| \right| ^ {r}\right) \\ = D _ {r} (G, G _ {*}). \\ \end{array}
$$
Case 2: If  $||(\Delta \mu, \Delta \Sigma)||^r + ||(\Delta \mu^*, \Delta \Sigma^*)||^r < ||(\mu, \Sigma) - (\mu^*, \Sigma^*)||^r$  and  $\lambda + \lambda^* \leq 1$ , then
$$
\begin{array}{l} W _ {r} ^ {r} (G, G _ {*}) = \lambda | | (\Delta \mu , \Delta \Sigma) | | ^ {r} + \lambda^ {*} | | (\Delta \mu^ {*}, \Delta \Sigma^ {*}) | | ^ {r} \\ = (\lambda^ {*} - \lambda) | | (\Delta \mu^ {*}, \Delta \Sigma^ {*}) | | ^ {r} + \lambda (| | (\Delta \mu , \Delta \Sigma) | | ^ {r} + | | (\Delta \mu^ {*}, \Delta \Sigma^ {*}) | | ^ {r}). \\ \end{array}
$$
From Cauchy-Schartz's inequality, we have  $||(\Delta \mu, \Delta \Sigma)||^r + ||(\Delta \mu^*, \Delta \Sigma^*)||^r \gtrsim ||(\mu, \Sigma) - (\mu^*, \Sigma^*)||^r$ . Therefore, under Case 2 we have  $||(\Delta \mu, \Delta \Sigma)||^r + ||(\Delta \mu^*, \Delta \Sigma^*)||^r \asymp ||(\mu, \Sigma) - (\mu^*, \Sigma^*)||^r$ , which directly implies that  $W_r^r(G, G_*) \asymp D_r(G, G_*)$ .
Case 3: If  $||(\Delta \mu, \Delta \Sigma)||^r + ||(\Delta \mu^*, \Delta \Sigma^*)||^r < ||(\mu, \Sigma) - (\mu^*, \Sigma^*)||^r$  and  $\lambda + \lambda^* > 1$ , then
$$
\begin{array}{l} W _ {r} ^ {r} (G, G _ {*}) = (1 - \lambda^ {*}) | | (\Delta \mu , \Delta \Sigma) | | ^ {r} + (1 - \lambda) | | (\Delta \mu^ {*}, \Delta \Sigma^ {*}) | | ^ {r} \\ + \quad (\lambda + \lambda^ {*} - 1) | | (\mu , \Sigma) - (\mu^ {*}, \Sigma^ {*}) | | ^ {r} \\ = (\lambda^ {*} - \lambda) | | (\Delta \mu^ {*}, \Delta \Sigma^ {*}) | | ^ {r} + (1 - \lambda^ {*}) (| | (\Delta \mu , \Delta \Sigma) | | ^ {r} + | | (\Delta \mu^ {*}, \Delta \Sigma^ {*}) | | ^ {r}) \\ + \quad (\lambda^ {*} + \lambda - 1) | | (\mu , \Sigma) - (\mu^ {*}, \Sigma^ {*}) | | ^ {r}. \\ \end{array}
$$
Since  $||(\Delta \mu, \Delta \Sigma)||^r + ||(\Delta \mu^*, \Delta \Sigma^*)||^r \asymp ||(\mu, \Sigma) - (\mu^*, \Sigma^*)||^r$ , we achieve
$$
(1 - \lambda^ {*}) (| | (\Delta \mu , \Delta \Sigma) | | ^ {r} \asymp (1 - \lambda^ {*}) | | (\mu , \Sigma) - (\mu^ {*}, \Sigma^ {*}) | | ^ {r}.
$$
Therefore, we also have  $W_r^r (G,G_*)\asymp D_r(G,G_*)$  under Case 3.
Combining the results from these cases, we reach the conclusion of the lemma.
F Discussion and Additional Experiments
F.1 Parameter Changes with the Sample Size
In statistics and machine learning, researchers often want to know how many samples are enough to achieve some pre-specified  $\epsilon$  error for the estimation of parameter  $\theta$  in the fitted model. In the language of probability, we want to find an inequality such as  $E\|\widehat{\theta}_n - \theta\| < C * rate(n)$ , where  $rate(n)$  is a decreasing sequence in  $n$  and  $C$  does not depend on  $n$ . Usually, in the parametric models, we have  $rate(n) = 1/\sqrt{n}$ , and therefore it takes  $C^2/\epsilon^2$  samples to achieve average  $\epsilon$  error in estimation. In complex models such as hierarchical models or the multivariate deviated model that we consider in this paper, difficulties arise because of the singularity and identifiability of the model. For example, in Eq. (2), if  $\lambda^* = 0$ , any pair of  $(\mu^*, \Sigma^*)$  yields the same model. Hence, when  $\lambda^* \approx 0$ , it should be harder to estimate  $(\mu^*, \Sigma^*)$ , and researchers may need more samples to have an accurate estimation for them. Notably, we have shown, for example in Theorem 4.1, the precise dependence of the convergence rate of  $(\mu^*, \Sigma^*)$  on the magnitude of  $\lambda^*$ . In particular, we have
$$
\mathbb {E} | \widehat {\lambda} _ {n} - \lambda^ {*} | \leq C \frac {\log n}{\sqrt {n}}, \quad \mathbb {E} \lambda^ {*} \left\| (\widehat {\mu} _ {n}, \widehat {\Sigma} _ {n}) - (\mu^ {*}, \Sigma^ {*}) \right\| \leq C \frac {\log n}{\sqrt {n}},
$$
where  $C$  is a constant that does not depend on  $\lambda^{*},\mu^{*},\Sigma^{*}$  and  $n$ . Hence, one can have a good estimation (with error  $\epsilon$ ) for  $\lambda^{*}$  with  $C^2 /\epsilon^2$  samples, while he needs  $C^2 /(\epsilon *\lambda^*)^2$  samples to achieve such a good estimation for  $(\mu^{*},\Sigma^{*})$ . The simulation studies in the next section will make this clearer.
(a) Rate of  $\widehat{\lambda}_n$
(b) Rate of  $\widehat{\mu}_n$
(c) Rate of  $\hat{\sigma}_n^2$
Figure 3: Case  $\lambda^{*} = 0.5 / n^{3 / 8}$
(b) Rate of  $\widehat{\mu}_n$
Figure 4: Case  $\lambda^{*} = 0.5 / n^{1 / 2}$
F.2 Additional Experiments for the Distinguishable Settings
We have seen in the main text that in two cases where  $\lambda^{*}$  is either fixed or decreasing with rate  $n^{-1/4}$ , the convergence rate of  $\lambda^{*}$  is  $C \times n^{-1/2}$ , where the constants  $C$  is the same for both cases. The convergence rate for  $(\mu^{*}, \Sigma^{*})$  is  $C \times n^{-1/4}$  for the latter case, which is much slower than the parametric rate in the former case. This phenomenon is quite rare for parametric models. We want to further bring readers' attention to two more extreme cases:
1.  $\lambda^{*} = 0.5 / n^{3 / 8}$  as  $n$  increases;
2.  $\lambda^{*} = 0.5 / n^{1 / 2}$  as  $n$  increases,
where we consider the same  $(\mu^{*},\Sigma^{*})$  with the experiments in the main text. The convergence rate for  $(\lambda ,\mu ,\sigma^2)$  in both cases in the log domain can be seen in Figure 3 and Figure 4. Hence, in all cases, the rate of convergence for  $\lambda^{*}$  is always of order  $n^{-1 / 2}$ , meanwhile, the rate for  $(\mu^{*},\Sigma^{*})$  becomes slower as  $\lambda^{*}$  tends to 0 faster. From the theoretical result, when  $\lambda^{*} = 0.5 / n^{3 / 8}$ , we expect the rate for  $(\mu^{*},\Sigma^{*})$  to be of order  $n^{-1 / 8}$ , which is demonstrated in Figure 3(b)&(c)). At the extreme case  $\lambda^{*} = 0.5 / n^{1 / 2}$ , it is even impossible to recover  $(\mu^{*},\Sigma^{*})$  as  $n\to \infty$  (cf. Figure 4(b)&(c)). This suggests practitioners collect more data when  $\hat{\lambda}_n$  is small to have a good estimate for  $(\mu^{*},\Sigma^{*})$ . Finally, in the case  $\hat{\lambda}_n$  is extremely small (of order  $n^{-1 / 2}$ ), we suggest not to report the estimated values  $(\widehat{\mu}_n,\widehat{\Sigma}_n)$ , as they are highly uncertain.
F.3 Non-distinguishable Settings
Finally, we consider the weakly identifiable and non-distinguishable setting here to demonstrate that the convergence rate for  $\lambda^*$  can be slower than the parametric rate when  $f$  is near  $h_0$ . Let both  $h_0$  and  $f$  belong to the location-scale Gaussian family.  $h_0(x) = f(x|0,1)$  and consider two cases of  $(\lambda^*,\mu^*,\Sigma^*)$ :
1.  $\lambda^{*} = 0.25, \mu^{*} = 0$ . are fixed and  $\sigma^{*} = 1 + n^{-1/8}$  as  $n$  increases;
2.  $\lambda^{*} = 0.25$ ,  $\sigma^{*} = 1$ . are fixed and  $\mu^{*} = n^{-1 / 8}$  as  $n$  increases;
Figure 5: Case 1:  $\mu^{*} = \mu_{0}$  and  $(\sigma^{*})^{2}\to \sigma_{n}^{2}$  in the rate  $n^{-1 / 8}$
(a) Rate of  $\widehat{\lambda}_n$
Figure 6: Case 2:  $\sigma^{*} = \sigma_{0}$  and  $\mu^{*}\rightarrow \mu_{0}$  in the rate  $n^{-1 / 8}$
Recall that we have proved that  $\left\{\|\Delta \mu^{*}\|^{4} + \|\Delta \Sigma^{*}\|^{2}\right\} |\widehat{\lambda}_{n} - \lambda^{*}| = \mathcal{O}(n^{-1 / 2})$  and  $\lambda^{*}(\|\Delta \mu^{*}\|^{2}+\|\Delta \widehat{\mu}_{n}\|^{2}+\|\Delta \Sigma^{*}\|+\|\Delta \widehat{\Sigma}_{n}\|)(\|\widehat{\mu}_{n}-\mu^{*}\|^{2}+\|\widehat{\Sigma}_{n}-\Sigma^{*}\|)=\mathcal{O}(n^{-1 / 2})$ , where there is a mismatch in the orders of convergence rates of the location and scale parameter. Notably, the rate of convergence for  $\lambda^{*}$  also depends on the rate  $\Delta \mu^{*}$  and  $\Delta (\sigma^{*})^{2}\to 0$ . The experiments do support this theoretical finding, where we have the rate for  $\lambda^{*}$  is  $\approx n^{-1 / 4}$  is the first case (as  $\left\|\Delta (\sigma^{*})^{2}\right\|^{2} = O(n^{-1 / 4}))$  and it does not converge in the second case  $\lambda^{*}$  is  $\approx n^{-1 / 4}$  is the first case (as  $\left\|\Delta \mu^{*}\right\|^{4} = O(n^{-1 / 2})$ ). The mismatch rate for  $\|\widehat{\mu}_n - \mu^*\|$  and  $\|\widehat{\sigma}_n^2 -(\sigma^*)^2\|$  can also be seen clearly in the second case, where the rate for the scale parameter is still of the parametric rate, whereas it is slower for the mean.