npbr (version 1.8)

dea_est: DEA, FDH and linearized FDH estimators.

Description

The function implements the empirical FDH (free disposal hull), LFDH (linearized FDH) and DEA (data envelopment analysis) frontier estimators.

Usage

dea_est(xtab, ytab, x, type = "dea")

Value

Returns a numeric vector with the same length as x.

Arguments

xtab

a numeric vector containing the observed inputs \(x_1,\ldots,x_n\).

ytab

a numeric vector of the same length as xtab containing the observed outputs \(y_1,\ldots,y_n\).

x

a numeric vector of evaluation points in which the estimator is to be computed.

type

a character equal to "dea", "fdh" or "lfdh".

Author

Hohsuk Noh.

Details

There are mainly two usual frontier estimation methods for preserving monotonicity: the free disposal hull (FDH) introduced by Deprins et al. (1984) and the data envelopment analysis (DEA) initiated by Farrell (1957). The FDH boundary is the lowest ``stair-case'' monotone curve covering all the data points $$\varphi_n(x):=\max\{y_i,\,i:x_i\leq x\}.$$ An improved version of this estimator, referred to as the linearized FDH (LFDH), is obtained by drawing the polygonal line smoothing the staircase FDH curve. It has been considered in Hall and Park (2002) and Jeong and Simar (2006). When the joint support of data is in addition convex, the DEA estimator is defined as the least concave majorant of the FDH frontier (see also Gijbels et al. (1999)). We employ the function DEA from the package Benchmarking to implement the function dea_est.

References

Bogetoft, P. and Otto, L. (2011), Benchmarking with DEA, SFA and R, Springer-Verlag.

Deprins, D., Simar, L. and H. Tulkens (1984). Measuring labor efficiency in post offices, in The performance of Public Enterprises: Concepts and Measurements (M. Marchand, P. Pestieau and H. Tulkens Eds), North-Holland, Amsterdam, 243--267.

Farrell, M.J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society: Series A, 120, 253--281.

Gijbels, I., Mammen, E., Park, B.U. and Simar, L. (1999). On estimation of monotone and concave frontier functions, Journal of American Statistical Association, 94, 220--228.

Hall, P. and Park, B.U. (2002). New methods for bias correction at endpoints and boundaries, Annals of Statistics, 30, 1460-1479.

Jeong, S.-O. and Simar, L. (2006). Linearly interpolated FDH efficiency score for nonconvex frontiers, Journal of Multivariate Analysis, 97, 2141--2161.

See Also

quad_spline_est, cub_spline_est.

Examples

data("green")
plot(OUTPUT~COST, data=green)
x <- seq(min(green$COST), max(green$COST), length.out=1001)
# We compute and represent the DEA, FDH and LFDH estimates
lines(x, dea_est(green$COST, green$OUTPUT, x, type="dea"),
 lty=4, lwd=4, col="cyan")  
lines(x, dea_est(green$COST, green$OUTPUT, x, type="fdh"),
 lty=1, lwd=4, col="green")
lines(x, dea_est(green$COST, green$OUTPUT, x, type="lfdh"), 
 lty=2, lwd=4, col="magenta")   
legend("topleft",legend=c("dea","fdh","lfdh"), 
 col=c("cyan","green","magenta"), lty=c(4,1,2), lwd=4)