state belongs to the class $\mathcal{R}(B,r)$, satisfies the upper bound (\[risk1\]) with $$\varphi_n^2 = \log(n)^{\frac{17}{3 r}} n^{-1}$$ obtained by taking $N(n) {\mathrel{\mathop:} = }{{\left( \frac{\log(n)}{2
        B} \right)}}^{\frac2r}$.

With the proposed $N(n)$ one checks that the bias (\[biais\]) is smaller than the variance (\[eq:variancesansbruit\]) which is bounded by a constant times ${\log(n)^{\frac{17}{3 r}}}{n^{-1}}$.

\[theo:peudebruit\] When $\frac12<\eta<1$, 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

-   For $r=2$, $$\varphi_n^2 = \log(n)^{ \frac{12\gamma+B}{3(4\gamma +B)}} 
                    n^{-\frac{B}{4\gamma+B}}$$ with $N(n) {\mathrel{\mathop:} = }\frac{\log(n)}{2(4\gamma+B)}\left( 1+\frac 23 \frac{\log(\log n)}{\log(n)}\right)$.

-   For $0<r<2$, $$\varphi_n^2 = \log(n)^{2-r/2}
                e^{-2B{N(n)}^{r/2}}$$ where $N(n)$ is the solution of the equation $8 \gamma N + 2 B
            N^{r/2} = \log(n)$.

In that case we have $N(n) = \frac{1}{8\gamma} \log(n) -
    \frac{2B}{(8\gamma)^{1+r/2}} \log(n)^{r/2} + o(\log(n)^{r/2})$.

When $r=2$, the proposed $N(n)$ ensures that the variance (\[eq:varpeudebruit\]) is equivalent to the bias (\[biais\]), which is bounded by a constant times $\log(n)^{\frac{12\gamma +B}{3(4\gamma +B)}}
    n^{-\frac{B}{4\gamma+B}}$.

When $0 < r < 2$, the proposed $N(n)$ makes the variance (\[eq:varpeudebruit\]) bounded by a constant times $e^{-2B{N(n)}^{r/2}}$, which is smaller than the bias, the latter being bounded by a constant times $N(n)^{2-r/2}
    e^{-2B{N(n)}^{r/2}}$.

The asymptotic expansion 