Integrated squared density curvature ($R(f^{\prime \prime })=\int (f^{\prime \prime }(x){)}^{2}dx$)

The integrated squared second derivative measures a smooth density's curvature in L2. It supplies the leading integrated squared bias constant of a second-order kernel density estimator. For nonzero second moment ${\mu }_2(K)$, the asymptotic mean integrated squared error has bias term $h^4{\mu }_2(K{)}^{2}R(f^{\prime \prime })/4$, so greater curvature calls for a smaller smoothing bandwidth.

For $f_a(x)=af(ax)$ with a positive, differentiation and change of variables give $R(f_a^{\prime \prime })=a^{5}R(f^{\prime \prime })$, while its standard deviation is $\sigma (f)/a$. Their product D is therefore invariant under scale changes, and translation also leaves it unchanged. Raw integrated squared curvature can be arbitrarily small under dilation even though this dimensionless shape measure is unchanged.