B /(1+\sqrt{B})^2$ there exists $c_2$, while in the case $0 < r < 2$, for any $\beta < B$ there exists $c_2$ and $n_0$ such that for $n \geq n_0$: $$\label{biaisbis}
        \sup_{\rho\in\mathcal{R}(B,r)}b_2^2(n) \leq
        c_2 N^{2} \delta^{4r-12} e^{-\frac{2\beta
          }{(2\delta)^r}-\frac 12
          \left(\frac{1}{\delta}-2\sqrt{N}\right)^2}.$$

Note that for $1/2 < \eta \leq 1$ we have $b_2(n)=0$ for all $0 < r
\leq 2$ ($\hat \rho^\eta_{j,k}$ is unbiased).

\[prop:5\] For $\hat{\rho}^\eta_{j,k}$ the estimator defined by (\[estrhojk\]), $$\begin{aligned}
        \label{eq:varbcpdebruit}
        \sup_{\rho\in\mathcal{R}(B,r)}\sigma^2(n) \leq c_3
        \frac{\delta N^{17/6}}{n} e^{\frac{2\gamma}{\delta^2}}
              &&\quad{\rm if \ } 0<\eta\leq 1/2\\
        \label{eq:varpeudebruit}
        \sup_{\rho\in\mathcal{R}(B,r)}\sigma^2(n) \leq c_3'
        \frac{N^{1/3}}{n}e^{8\gamma N}&&\quad{\rm if \ }1/2<\eta<1 \\
        \label{eq:variancesansbruit}
        \sup_{\rho\in\mathcal{R}(B,r)}\sigma^2(n)
        \leq c_3'' \frac{N^{\frac{17}{6}}}{n}&& \quad{\rm if \ } \eta = 1
    \end{aligned}$$ where $c_3$, $c_3'$ are positive constants depending on $\eta$.

We measure the accuracy of $\hat{\rho}^\eta_{j,k}$ by the maximal risk over the class $\mathcal{R}(B,r)$ $$\begin{aligned}
    \label{risk1}
    \limsup_{n\to\infty}
    \sup_{\rho\in\mathcal{R}(B,r)}\varphi_n^{-2}E\left(\sum_{j,k=0}^\infty
        \left|\hat{\rho}^\eta_{j,k}- \rho_{j,k}\right|^2\right)\leq
    C_0.\end{aligned}$$ where $C_0$ is a positive constant and $\varphi_n^{2}$ is a sequence which tends to 0 when $n\rightarrow\infty$ and it is the rate of convergence. Cases $\eta = 1$ (no noise), $\frac12<\eta<1$ (weak noise) and $0<\eta\leq\frac12$ (strong noise) are studied respectively in Theorems \[theo:sansbruit\], \[theo:peudebruit\] and \[theo:bcpdebruit\].

\[theo:sansbruit\] When $\eta = 1$, the estimator defined in (\[estrhojk\]) for the model (\[noisy.data\]), where the unknown 