quantreg (version 4.77)

rearrange: Rearrangement

Description

Monotonize a step function by rearrangement

Usage

rearrange(f,xmin,xmax)

Arguments

f

object of class stepfun

xmin

minimum of the support of the rearranged f

xmax

maximum of the support of the rearranged f

Value

• Produces transformed stepfunction that is monotonic increasing.

Details

Given a stepfunction $Q(u)$, not necessarily monotone, let $F(y) = \int { Q(u) \le y } du$ denote the associated cdf obtained by randomly evaluating $Q$ at $U \sim U[0,1]$. The rearranged version of $Q$ is $\tilde Q (u) = \inf { u: F(y) \ge u }. The rearranged function inherits the right or left continuity of original stepfunction.$

References

Chernozhukov, V., I. Fernandez-Val, and A. Galichon, (2006) Quantile and Probability Curves without Crossing, Econometrica, forthcoming.

Chernozhukov, V., I. Fernandez-Val, and A. Galichon, (2009) Improving Estimates of Monotone Functions by Rearrangement, Biometrika, 96, 559--575.

Hardy, G.H., J.E. Littlewood, and G. Polya (1934) Inequalities, Cambridge U. Press.

See Also

rq rearrange

Examples

data(engel)
z <- rq(foodexp ~ income, tau = -1,data =engel)
zp <- predict(z,newdata=list(income=quantile(engel$income,.03)),stepfun = TRUE)
plot(zp,do.points = FALSE, xlab = expression(tau),
        ylab = expression(Q ( tau )), main="Engel Food Expenditure Quantiles")
plot(rearrange(zp),do.points = FALSE, add=TRUE,col.h="red",col.v="red")
legend(.6,300,c("Before Rearrangement","After Rearrangement"),lty=1,col=c("black","red"))