Kernel Estimation 核估计

Kernel Estimation 核估计

1 Kernel Function

1.1 General Kernel Functions

• Triangular

\[\kappa(u) = (1 - |u|), \quad |u| < 1 \]

• Epanechnikov (parabolic)

\[\kappa(u) = \frac{3}{4} \left(1 - u^2 \right), \quad |u| < 1 \]

• Quartic (biweight)

\[\kappa(u) = \frac{15}{16} \left(1 - u^2 \right)^2, \quad |u| < 1 \]

• Triweight

\[\kappa(u) = \frac{35}{32} \left(1 - u^2 \right)^3, \quad |u| < 1 \]

• Tricube

\[\kappa(u) = \frac{70}{81} (1 - |u|^3 )^3, \quad |u| < 1 \]

• Gaussian

\[\kappa(u) = {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}u^{2}}, \quad |u| < 1 \]

References

Kernel (statistics), wikipedia, site