1.7 Influence Function Examples

  • Total: \[ \begin{aligned} IT_k(M)&=lim_{t\rightarrow 0}\frac{T(M+t\delta_k)-T(M)}{t}\\ &=lim_{t\rightarrow 0}\frac{\int y.d(M+t\delta_k)-\int y.dM}{t}\\ &=lim_{t\rightarrow 0}\frac{\int yd(t\delta_k)}{t}=y_k \end{aligned} \]

  • Ratio of two totals: \[ \begin{aligned} IR_k(M)&=I\left(\frac{U}{V}\right)_k(M)=\frac{V(M)\times IU_k(M)-U(M)\times IV_k(M)}{V(M)^2}\\ &=\frac{X y_k-Y x_k}{X^2}=\frac{1}{X}(y_k-Rx_k) \end{aligned} \]