Bias, variance, and signal-to-noise regimes of the isotropic risk curve

Summing the two components in [F1] gives the total risk $R(\gamma )={\sigma }^2\gamma \mathrm{/}(1-\gamma )$ for $0<\gamma <1$ and $R(\gamma )=r^2(1-1\mathrm{/}\gamma )+{\sigma }^2\mathrm{/}(\gamma -1)$ for $\gamma >1$. Differentiating the first expression gives $R^{\mathrm{\prime }}(\gamma )={\sigma }^2\mathrm{/}(1-\gamma {)}^2$, so it is strictly increasing for $\sigma >0$ and is identically zero for $\sigma =0$.

For $\gamma >1$, $R^{\mathrm{\prime }}(\gamma )=\frac{r^2}{{\gamma }^2}-\frac{{\sigma }^2}{(\gamma -1{)}^2}\mathrm{.}$ If $\sigma >0$ and $r\le \sigma$, then $r\mathrm{/}\gamma \le \sigma \mathrm{/}\gamma <\sigma \mathrm{/}(\gamma -1)$, so $R^{\mathrm{\prime }}<0$. If $r>\sigma >0$, the derivative vanishes exactly when $r(\gamma -1)=\sigma \gamma$, namely at ${\gamma }_{\ast }=r\mathrm{/}(r-\sigma )>1$; it is negative before this point and positive after it, giving the unique minimum. If $\sigma =0$, the derivative is $r^2\mathrm{/}{\gamma }^2$, which is positive for $r>0$ and zero for $r=0$.

The limits at $\gamma \downarrow 0$, $\gamma \uparrow 1$, and $\gamma \to \mathrm{\infty }$ follow directly from the formulas in [F1]. In particular, the singular terms at $\gamma =1$ vanish when $\sigma =0$ and diverge when $\sigma >0$. For $\sigma >0$ every finite-domain risk value is positive while the underparameterized branch tends to zero as $\gamma \downarrow 0$, proving that its global infimum is $0$ and is not attained. When $\sigma =0$, the underparameterized branch equals $0$ identically.