Table 5 Examples of symmetrical and asymmetrical kernel functions [46]

Epanechnikov

\(K\left( t\right) =\left\{ \begin{array}{lll} \frac{3}{4\sqrt{5}}\left( 1-\frac{1}{5}t \right) ^2,&{} \left| t \right| < \sqrt{5} &{} \\ 0, &{} \left| t \right| \ge \sqrt{5}. \end{array}\right.\)

Gamma 1

\(K\left( x,a; t \right) = \frac{t^{x/a}e^{-t/a}}{a^{{x/a}+1}\Gamma \left( {x/a}+1 \right) }.\)

Biweight

\(K\left( t\right) =\left\{ \begin{array}{lll} \frac{15}{16}\left( 1-t^2 \right) ^2,&{} \left| t \right| < 1 &{} \\ 0, &{} \left| t \right| \ge 1. \end{array}\right.\)

Gamma 2

\(\begin{array}{lll} &{} K\left( \rho _a\left( x \right) ,a; t \right) = \frac{t^{\rho _a\left( x \right) -1}e^{-t/a}}{a^{\rho _a\left( x \right) }\Gamma \left( \rho _a\left( x \right) \right) },\\ {} &{} \rho _a\left( x \right) = \left\{ \begin{array}{lll}x/a, &{} x \ge 2a \\ \frac{1}{4}\left( x/a\right) ^2+1, &{} x\in \left[ 0,2a \right) . \end{array}\right. \end{array}\)

Gaussian

\(K\left( t \right) = \frac{1}{\sqrt{2\pi } } e^{- \frac{t^{2}}{2} }\)

Reciprocal inverse Gaussian

\(K\left( x,a; t \right) = \frac{1}{\sqrt{2 \pi a t} }e^{-\frac{x-a}{2a}\left( \frac{t}{x-a}-2+\frac{x-a}{t}\right) }.\)

Rectangular

\(K\left( t\right) =\left\{ \begin{array}{ll}\frac{1}{2}, &{} \left| t \right| < 1\\ 0, &{} \left| t \right| \ge 1.\end{array}\right.\)

Lognormal

\(K\left( x,a; t \right) = \frac{1}{\sqrt{8 \pi \textrm{ln} \left( 1+a \right) t} }e^{- \frac{ \left( \textrm{ln}t-\textrm{ln}x \right) ^{2} }{8\textrm{ln}(1+a)} }.\)