Use the fact that the pointwise bias of the kernel density estimator with bandwidth \(h\) is given by

\[\operatorname{Bias}\left[\tilde{f}_{n, h}(x)ight]=\frac{1}{2} h^{2} f^{\prime \prime}(x) \sigma_{k}^{2}+O\left(h^{4}ight),\]

as \(h ightarrow 0\) to prove that the square bias is given by

\[\operatorname{Bias}^{2}\left[\tilde{f}_{n, h}(x)ight]=\frac{1}{4} h^{4}\left[f^{\prime \prime}(x)^{\prime 2} \sigma_{k}^{4}+O\left(h^{6}ight) .ight.\]