is unchanged, whereas the variance part is infinitely larger than the deconvolution variance in . As the bias is dominating over the variance in this setup, we can still choose a suitable sequence $s_n$ so that the same bandwidth is optimal associated to the same optimal rate, provided that $W_\rho$ decreases fast enough asymptotically. The following proposition gives upper bounds for the three components of the $\mathbb{L}_2$ risk uniformly over the class $\mathcal R(B,r)$.

\[bias.varL2\] Let $(Y_\ell,\Phi_\ell),\, \ell = 1,\ldots,n$ be i.i.d. data coming from the model  ($\ref{noisy.data}$) and let $\widehat
    W_{h}^\eta$ be an estimator (with $h \to 0$ as $n \to \infty$) of the underlying Wigner function $W_\rho$. We suppose $W_\rho$ lies in the class $\mathcal R(B,r)$, with $B>0$ and $0<r\leq 2$. Then, for $s_n \to \infty$ as $n \to \infty$ and $n$ large enough, $$\begin{aligned}
        \sup_{\rho \in \mathcal R(B,r)}
        \| W^\rho \|^2_{\bar D(s_n)}
        &\leq &
        C_1 s_n^{10-3r} e^{-2\beta s_n^r}   ,\\
[2mm]
        \sup_{\rho \in \mathcal R(B,r)}\left\|E[\widehat
            W^\eta_{h}] -W_\rho \right\|^2_{D(s_n)}
        & \leq &
        C_{2 } h^{3r-10} e^{- \frac{2^{1-r} \beta}{ h^r}}   \\
[2mm]
        \sup_{\rho \in \mathcal R(B,r)}
        E\left[\left\|\widehat W_{h,n}^\eta - E
                \left[\widehat W_{h,n}^\eta \right]\right\|_{D(s_n)}^2 \right]
        & \leq & C_3
        \frac{s_n^2}{ n h} \exp \left(\frac{2 \gamma}{h^2} \right),
    \end{aligned}$$ where $\beta < B 