DepthProc (version 1.0.7)

deepReg2d: Simple deepest regression method.

Description

This function calculates deepest regression estimator for simple regression.

Usage

deepReg2d(x, y)

Arguments

x

Independent variable.

y

Dependent variable.

Details

Function originates from an original algorithm proposed by Rousseeuw and Hubert. Let \( {Z}^{n}={ (x_1,y_1),...,(x_n,y_n)} {\subset {R}^{d} }\) denotes a sample considered from a following semiparametric model: \( {{y}_{l}}={{a}_{0}}+{{a}_{1}}{{x}_{1l}}+...+{{a}_{(d-1)l}}{{x}_{(d-1)l}}+{{\varepsilon }_{l}}\), \(l=1,...,n\), we calculate a depth of a fit \( \alpha=(a_{0},...,a_{d-1}) \) as \( RD(\alpha ,{{Z}^{n}})={u\ne 0}{{\min }}\,\sharp{l: \frac{{{r}_{l}}(\alpha )}{{{u}^{T}}{{x}_{l}}}<0,l=1,...,n}\), where \( r(\cdot ) \) denotes the regression residual, \( \alpha=(a_{0},...,a_{d-1}) \) , \( {u}^{T}{x}_{l}\ne 0 \) . The deepest regression estimator \( DR(\alpha,{{Z}^{n}}) \) is defined as

\( DR(\alpha ,{{Z}^{n}})={\alpha \ne 0}{{\arg \max }}\,RD(\alpha ,{{Z}^{n}})\)

References

Rousseeuw J.P., Hubert M. (1998), Regression Depth, Journal of The American Statistical Association, vol.94.

Examples

data(pension)
 plot(pension)
 abline(lm(Reserves~Income,data = pension), lty = 3, lwd = 2) #lm
 abline(deepReg2d(pension[,1],pension[,2]), lwd = 2) #deepreg2d
 #EXAMPLE 2
 data(under5.mort)
 data(inf.mort)
 data(maesles.imm)
 data2011=na.omit(cbind(under5.mort[,22],inf.mort[,22],maesles.imm[,22]))
 x<-data2011[,3]
 y<-data2011[,2]
 plot(x,y,cex=1.2, ylab="infant mortality rate per 1000 live birth",
 xlab="against masles immunized #'  percentage",
 main='Projection Depth Trimmed vs. LS regressions')
 abline(lm(x~y,data = pension), lwd = 2, col='black') #lm
 abline(deepReg2d (x,y), lwd = 2,col='red') #trimmed reg
 legend("bottomleft",c("LS","DeepReg"),fill=c("black","red"),cex=1.4,bty="n")