of $N(n)$ is a standard consequence of its definition by the equation $8 \gamma N + 2 B N^{r/2} =
    \log(n)$.

\[theo:bcpdebruit\] When $0 < \eta \leq \frac12$, the estimator defined in (\[estrhojk\]) for the model (\[noisy.data\]), where the unknown state belongs to the class $\mathcal{R}(B,r)$, satisfies the upper bound (\[risk1\]) with $$\varphi_n^2 = N^{2-r/2}e^{-2 B N^{r/2}}$$ where $N$ and $\delta$ are solutions of the system $$\left\{ \begin{array}{ll}
            \frac{2 \beta}{(2\delta)^r} + \frac12
            (\frac1\delta-2\sqrt{N})^2 + \frac{2 \gamma}{\delta^2}
            = \log(n)\\
            \frac{2\beta}{(2\delta)^r} + \frac12
            (\frac1\delta-2\sqrt{N})^2 - 2B N^{r/2} = (\log\log(n))^2
                                       \end{array} 
                               \right. 
        \label{eq:7}$$ for arbitrary $\beta < B$ in the case $0 < r < 2$ or $$\left\{ \begin{array}{ll}
            \frac{\beta + 4 \gamma}{2 \delta^2} + \frac12
            (\frac1\delta - 2\sqrt{N})^2 - \frac53 \log(N) = \log(n) \\
            \frac{\beta}{2\delta^2} + \frac12 (\frac1\delta -
            2\sqrt{N})^2 - 2BN - 3 \log(N) = 0
                                       \end{array} 
                               \right. 
       \label{eq:8}$$ with $\beta {\mathrel{\mathop:} = }\frac{B}{(1+\sqrt{B})^2}$ in the case $r = 2$.

Theses bounds are optimal in the sense that (\[eq:7\]) and (\[eq:8\]) are obtained by minimizing the sum of the bounds (\[biais\]), (\[biaisbis\]) and (\[eq:varbcpdebruit\]).

We use the standard notations $a(n) \sim b(n)$ if $\frac{a(n)}{b(n)} \to 1$ and $a(n) \approx b(n)$ if there exists a constant $M < \infty$ such that $\frac{1}{M} 