Package 'frontiles' reference manual

Title:	Partial Frontier Efficiency Analysis
Description:	It calculates the alpha-quantile proposed by Daouia and Simar (2007) <doi:10.1016/j.jeconom.2006.07.002> and order-m efficiency score in multi-dimension proposed by Daouia and Gijbels (2011) <doi:10.1016/j.jeconom.2010.12.002> and computes several summaries and representation of the associated frontiers in 2d and 3d.
Authors:	Thibault Laurent [aut, cre, cph], Abdelaati Daouia [aut]
Maintainer:	Thibault Laurent <[email protected]>
License:	GPL (>= 2)
Version:	1.3.1
Built:	2024-11-09 06:12:57 UTC
Source:	CRAN

alpha-quantile efficiency score and 2d/3d representation of frontier

Description

It calculates the alpha-quantile efficiency score and order-m score in multi-dimension and gives representation of alpha-quantile efficiency frontier. This work was supported by the agence nationale de la recherche through the EPI project (ANR-08-BLAN-0106-01)

Details

Package:	frontiles
Type:	Package
Version:	1.2
Date:	2023-04-01
License:	GPL Vesion 2 or later
LazyLoad:	yes

Author(s)

Abdelaati Daouia and Thibault Laurent

Maintainer: <[email protected]>

References

Daouia, A. and L. Simar (2007), Nonparametric efficiency analysis: A multivariate conditional quantile approach, Journal of Econometrics 140, 375-400.

Examples

data(spain)
xobs=as.matrix(spain[,3:4])
yobs=as.matrix(spain[,1])
score.ref.1<-alphascore(xobs,yobs)
data(spain)
xobs=as.matrix(spain[,3:4])
yobs=as.matrix(spain[,1])
score.ref.1<-alphascore(xobs,yobs)

Representation of alpha-quantile efficiency frontier for 1 output and 1 input

Description

Representation of the alpha-quantile efficiency frontier (output, input or hyperbolic direction) for a set of reference points (xobs, yobs) in 2D (1 output and 1 input).

Usage

alphafrontier.2d(xobs, yobs, type="output", alpha=0.95, add=FALSE, 
confidence=FALSE, shade=FALSE,...)
alphafrontier.2d(xobs, yobs, type="output", alpha=0.95, add=FALSE, 
confidence=FALSE, shade=FALSE,...)

Arguments

`xobs`	a matrix of size $n_1 \times 1$ , input of sample points
`yobs`	a matrix of size $n_1 \times 1$ , output of sample points
`type`	a direction to choose among "output", "input" and "hyper"
`alpha`	a scalar between 0 and 1
`add`	a boolean with TRUE for keeping the active device
`confidence`	a boolean for representing a confidence interval
`shade`	a boolean for shading the confidence interval
`...`	usual options for ploting the frontier, lty, col, etc.

Details

Actually, there is no confidence interval when type="hyper". If type="input" and confidence=TRUE, the y-axis is permuted with the x-axis

Value

No return value, used for plotting the alpha-quantile efficiency frontier for 1 output and 1 input

Author(s)

Abdelaati Daouia and Thibault Laurent

References

Daouia, A. and L. Simar (2007), Nonparametric efficiency analysis: A multivariate conditional quantile approach, Journal of Econometrics 140, 375-400.

Examples

# 1st example
data(spain)
plot(y ~ x2, data = spain)
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  alpha = 0.95, col = 'red', lty = 2, add = TRUE)
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "input",
  alpha = 0.95, col = 'royalblue', lty = 3, add = TRUE)
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "hyper",
  alpha = 0.95, col = 'green', lty = 4, add = TRUE)
legend("topleft", title = "alpha-quantile frontier; alpha=0.95",
  legend = c("output direction", "input direction", "hyper direction"), 
  lty = 2:4, col = c("red", "royalblue", "green"))

# 2nd example
plot(y ~ x2, data = spain)
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y),
  type = "output", alpha = 1, add = TRUE)
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  alpha = 0.95, col = 'blue', lty = 2, add = TRUE)
ordermfrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  m = 30, col = 'green', lty = 3, add = TRUE)
legend("topleft", title = "output direction", legend = c("FDH", "alpha=0.95", "m=30"),
  lty = 1:3, col = c("black", "royalblue", "green"))

# 3rd example
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  confidence = TRUE, shade = TRUE, alpha = 0.98)
title("Alpha-quantile frontier with alpha=0.98 and its confidence interval")
# 1st example
data(spain)
plot(y ~ x2, data = spain)
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  alpha = 0.95, col = 'red', lty = 2, add = TRUE)
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "input",
  alpha = 0.95, col = 'royalblue', lty = 3, add = TRUE)
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "hyper",
  alpha = 0.95, col = 'green', lty = 4, add = TRUE)
legend("topleft", title = "alpha-quantile frontier; alpha=0.95",
  legend = c("output direction", "input direction", "hyper direction"), 
  lty = 2:4, col = c("red", "royalblue", "green"))

# 2nd example
plot(y ~ x2, data = spain)
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y),
  type = "output", alpha = 1, add = TRUE)
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  alpha = 0.95, col = 'blue', lty = 2, add = TRUE)
ordermfrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  m = 30, col = 'green', lty = 3, add = TRUE)
legend("topleft", title = "output direction", legend = c("FDH", "alpha=0.95", "m=30"),
  lty = 1:3, col = c("black", "royalblue", "green"))

# 3rd example
alphafrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  confidence = TRUE, shade = TRUE, alpha = 0.98)
title("Alpha-quantile frontier with alpha=0.98 and its confidence interval")

3d representation of alpha-quantile frontier in the case of 2 input and 1 output.

Description

3d representation of alpha-quantile frontier for a set of reference points (xobs,yobs) in the case of two input and one output. No representation yet for hyperbolic direction.

Usage

alphafrontier.3d(xobs, yobs, type="output",alpha=0.95, digits=4, 
box.leg=TRUE, palette=heat_hcl, rgl=FALSE, n.class=NULL,  ...)
alphafrontier.3d(xobs, yobs, type="output",alpha=0.95, digits=4, 
box.leg=TRUE, palette=heat_hcl, rgl=FALSE, n.class=NULL,  ...)

Arguments

`xobs`	a matrix of size $n_1 \times 2$ , input of reference points
`yobs`	a matrix of size $n_1 \times 1$ , output of reference points
`type`	a character, "output" or "input" direction
`alpha`	a scalar
`digits`	a precision parameter to compute the alpha-frontier
`box.leg`	representation of a legend-box on the plot with the values of frontier
`palette`	function to use for colors in case where option `col` has not been called. See package colorspace for more informations.
`rgl`	a boolean, for output direction, representation of the graphic in 3d if TRUE
`n.class`	a numeric, for output direction, the number of class
`...`	usual parameters of function `plot`. Use the parameter `asp` to modify the scale of window...

Details

In "input" direction: You choose a value of output in the legend box, the efficiency-frontier of the input is represented with the corresponding color on the 2d graphic. In "output" direction : For calculate the alpha-quantile efficiency output frontier everywhere, we have constructed a grid of size $n \times n$ by drawing vertical and horizontal lines which intersect each reference observations. Then, we have calculated for each cell $C_k$ $k=1,...,n^2$ the prediction of the alpha-quantile output efficiency frontier which correspond to the $\alpha^{th}$ elements of the suite $\{y_j\}_{j=1,...,n_k}$ where reference observations j, $j=1,...,n_k$ verify

$\ x^1_j\leq\inf_{(x^1,x^2) \in C_k}x^1$

and $x^2\leq\inf_{(x_1,x_2) \in C_k}x^2$ .
We propose to use an algorithm which filled up cells with colors depending on the values taken by the alpha-quantile output efficiency frontier. The algorithm attach the row and vary the column as the folling figure can show it.

Value

no values

Note

The algorithm used is certainly not optimized. For a data set of 61 observations, the function necessits 15.17s on an Optiplex GX745 2 duo 2.13GHz under Windows Vista and probably bugs beyond a certain number of observation

Author(s)

Abdelaati Daouia and Thibault Laurent

Examples

data(spain)
xyn <- cbind(spain[, 3:4],spain[, 1])
xtab <- as.matrix(xyn[, c(1,2)])
ytab <- matrix(xyn[, 3])

# representation in 2-d

op <- par(no.readonly = TRUE) # the whole list of settable par's.
alphafrontier.3d(xtab, ytab, type = "output", alpha = 0.6, xlab = "input 1",
 ylab = "input 2", main = "blabla") 
points(xtab, pch = 16)
par(op)
data(spain)
xyn <- cbind(spain[, 3:4],spain[, 1])
xtab <- as.matrix(xyn[, c(1,2)])
ytab <- matrix(xyn[, 3])

# representation in 2-d

op <- par(no.readonly = TRUE) # the whole list of settable par's.
alphafrontier.3d(xtab, ytab, type = "output", alpha = 0.6, xlab = "input 1",
 ylab = "input 2", main = "blabla") 
points(xtab, pch = 16)
par(op)

Calculates alpha-quantile efficiency score

Description

Calculates alpha-quantile efficiency score (output, input and hyperbolic direction) for a set of evaluation points (xeval, yeval) depending on reference points (xobs, yobs).

Usage

alphascore(xobs, yobs, xeval=xobs, yeval=yobs, alpha=0.95)
alphascore(xobs, yobs, xeval=xobs, yeval=yobs, alpha=0.95)

Arguments

`xobs`	a matrix of size $n_1 \times p$ , input of sample points
`yobs`	a matrix of size $n_1 \times q$ , output of sample points
`xeval`	a matrix of size $n_2 \times p$ , input of assessment points
`yeval`	a matrix of size $n_2 \times q$ , output of assessment points
`alpha`	a scalar

Details

A score between 0 and 1 means that DMU is inefficient. If DMU greater than 1, DMU is super-efficient.

Value

a data.frame object with the alpha-quantile efficiency score in:

`output`	output direction
`input`	input direction
`hyper`	hyperbolic direction

Author(s)

Abdelaati Daouia and Thibault Laurent

References

Daouia, A. and L. Simar (2007), Nonparametric efficiency analysis: A multivariate conditional quantile approach, Journal of Econometrics 140, 375-400.

Examples

# 1st example
data(spain)
res.alqf <- alphascore(xobs = as.matrix(spain[, c(2, 3, 4)]),
               yobs = as.matrix(spain[, 1]), alpha = 0.8)

# 2nd example
data(burposte)
bur.samp <- burposte[which(burposte$xinput < 50000), ]
ind.samp <- sample(nrow(bur.samp), 500)
xeval <- as.matrix(bur.samp[ind.samp[1:100], 2])
yeval <- as.matrix(bur.samp[ind.samp[1:100], 3])
xobs <- as.matrix(bur.samp[ind.samp[101:500], 2])
yobs <- as.matrix(bur.samp[ind.samp[101:500], 3])

alphafrontier.2d(xobs, yobs, alpha = 0.95)
points(xeval, yeval, pch = 16, col = 'red')
text(xeval, yeval, as.character(1:100), adj = 2, cex = 0.8)
score.new.0.95 <- alphascore(xobs, yobs, xeval, yeval, alpha = 0.95)
# 1st example
data(spain)
res.alqf <- alphascore(xobs = as.matrix(spain[, c(2, 3, 4)]),
               yobs = as.matrix(spain[, 1]), alpha = 0.8)

# 2nd example
data(burposte)
bur.samp <- burposte[which(burposte$xinput < 50000), ]
ind.samp <- sample(nrow(bur.samp), 500)
xeval <- as.matrix(bur.samp[ind.samp[1:100], 2])
yeval <- as.matrix(bur.samp[ind.samp[1:100], 3])
xobs <- as.matrix(bur.samp[ind.samp[101:500], 2])
yobs <- as.matrix(bur.samp[ind.samp[101:500], 3])

alphafrontier.2d(xobs, yobs, alpha = 0.95)
points(xeval, yeval, pch = 16, col = 'red')
text(xeval, yeval, as.character(1:100), adj = 2, cex = 0.8)
score.new.0.95 <- alphascore(xobs, yobs, xeval, yeval, alpha = 0.95)

Burposte data

Description

Burposte data

Usage

data(burposte)data(burposte)

Format

A data frame with 9521 observations on the following 3 variables.

ident: a numeric vector
xinput: a numeric vector
yprod: a numeric vector

References

Cazals C., Florens J.-P., Simar L. (2002), Nonparametric frontier estimation: a robust approach, Journal of Econometrics 106, 1-25.

Examples

data(burposte)
data(burposte)

Representation of m-order efficiency frontier for 1 output and 1 input

Description

Representation of the m-order efficiency score (output, input or hyperbolic direction) for a set of reference points (xobs, yobs) in 2D (1 output and 1 input).

Usage

ordermfrontier.2d(xobs, yobs, type = "output", m = 30, add = FALSE,
     confidence = FALSE, shade = FALSE,...)
ordermfrontier.2d(xobs, yobs, type = "output", m = 30, add = FALSE,
     confidence = FALSE, shade = FALSE,...)

Arguments

`xobs`	a matrix of size $n_1 \times 1$ , input of sample points
`yobs`	a matrix of size $n_1 \times 1$ , output of sample points
`type`	a direction to choose among "output", "input" and "hyper"
`m`	an integer
`add`	a boolean with TRUE for keeping the active device
`confidence`	a boolean for representing a confidence interval
`shade`	a boolean for shading the confidence interval
`...`	usual options for ploting the frontier, lty, col, etc.

Details

Actually, there is no confidence interval when type="hyper". If type="input" and confidence=TRUE, the y-axis is permuted with the x-axis

Value

No return value, used for plotting the m-order efficiency frontier for 1 output and 1 input

Author(s)

Abdelaati Daouia and Thibault Laurent

References

Daouia, A. and L. Simar (2007), Nonparametric efficiency analysis: A multivariate conditional quantile approach, Journal of Econometrics 140, 375-400.

Examples

# 1st example
data(spain)
plot(y ~ x2, data = spain)
ordermfrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  m = 30, col = 'red', lty = 2, add = TRUE)
ordermfrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "input", m = 30,
  col = 'royalblue', lty = 3, add = TRUE)
ordermfrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "hyper", m = 30,
  col = 'green', lty = 4, add = TRUE)
legend("topleft", legend = c("output direction", "input direction", "hyper direction"),
  lty = 2:4, col = c("red", "royalblue", "green"))

# 2nd example
ordermfrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  confidence = TRUE, shade = TRUE, m = 30)
# 1st example
data(spain)
plot(y ~ x2, data = spain)
ordermfrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  m = 30, col = 'red', lty = 2, add = TRUE)
ordermfrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "input", m = 30,
  col = 'royalblue', lty = 3, add = TRUE)
ordermfrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "hyper", m = 30,
  col = 'green', lty = 4, add = TRUE)
legend("topleft", legend = c("output direction", "input direction", "hyper direction"),
  lty = 2:4, col = c("red", "royalblue", "green"))

# 2nd example
ordermfrontier.2d(as.matrix(spain$x2), as.matrix(spain$y), type = "output",
  confidence = TRUE, shade = TRUE, m = 30)

Calculates order-m efficiency score with asymptotic formula

Description

Calculates order-m efficiency score (output, input and hyperbolic direction) for a set of assessment points (xeval, yeval) depending on sample points (xobs, yobs), using the formulas of Daouia and Gijbels (2011).

Usage

ordermscore(xobs, yobs, xeval=xobs, yeval=yobs, m=30)
ordermscore(xobs, yobs, xeval=xobs, yeval=yobs, m=30)

Arguments

`xobs`	a matrix of size $n_1 \times p$ , input of sample points
`yobs`	a matrix of size $n_1 \times q$ , output of sample points
`xeval`	a matrix of size $n_2 \times p$ , input of assessment points
`yeval`	a matrix of size $n_2 \times q$ , output of assessment points
`m`	an integer

Details

A score between 0 and 1 means that DMU is inefficient. If DMU greater than 1, DMU is super-efficient. The asymptotic formula of the order-m score are given in Daouia and Gijbels (2011).

Value

a data.frame object with the order-m efficiency score in:

`output`	output direction
`input`	input direction
`hyper`	hyperbolic direction

Author(s)

Abdelaati Daouia and Thibault Laurent

References

Daouia and Gijbels (2011), Robustness and inference in nonparametric partial-frontier modeling, Journal of Econometrics.

Examples

# 1st example
data(spain)
score.orderm<-ordermscore(xobs = as.matrix(spain[, c(2, 3, 4)]),
                          yobs = as.matrix(spain[, 1]))

# 2nd example
data(burposte)
ind.samp <- sample(nrow(burposte), 500)
xobs <- as.matrix(burposte[ind.samp[1:100], 2])
yobs <- as.matrix(burposte[ind.samp[1:100], 3])
xeval <- as.matrix(burposte[ind.samp[101:500],2])
yeval <- as.matrix(burposte[ind.samp[101:500], 3])
score.orderm.2 <- ordermscore(xobs, yobs, xeval, yeval)
# 1st example
data(spain)
score.orderm<-ordermscore(xobs = as.matrix(spain[, c(2, 3, 4)]),
                          yobs = as.matrix(spain[, 1]))

# 2nd example
data(burposte)
ind.samp <- sample(nrow(burposte), 500)
xobs <- as.matrix(burposte[ind.samp[1:100], 2])
yobs <- as.matrix(burposte[ind.samp[1:100], 3])
xeval <- as.matrix(burposte[ind.samp[101:500],2])
yeval <- as.matrix(burposte[ind.samp[101:500], 3])
score.orderm.2 <- ordermscore(xobs, yobs, xeval, yeval)

Calculates order-m efficiency score with bootstrap algorithm

Description

Calculates order-m efficiency score (output, input and hyperbolic direction) for a set of assessment points (xeval, yeval) depending on sample points (xobs, yobs), using the initial algorithm of Cazals et al. (2002).

Usage

ordermscore.boot(xobs, yobs, xeval=xobs, yeval=yobs, m=30, B=200, m.move=FALSE)
ordermscore.boot(xobs, yobs, xeval=xobs, yeval=yobs, m=30, B=200, m.move=FALSE)

Arguments

`xobs`	a matrix of size $n_1 \times p$ , input of sample points
`yobs`	a matrix of size $n_1 \times q$ , output of sample points
`xeval`	a matrix of size $n_2 \times p$ , input of assessment points
`yeval`	a matrix of size $n_2 \times q$ , output of assessment points
`m`	an integer, the number of selected firms
`B`	an integer, the number of replication
`m.move`	a boolean, to choose different values of m

Details

This function computes the algorithm initially proposed by Cazals et al. (2002). If m.move=TRUE, different values of m are given as suggested by Daouia et al (2009).

Value

a data.frame object with the average mean order-m efficiency score and standard deviation associated:

`output`	output direction
`output`	output direction
`input`	input direction
`input`	input direction
`hyper`	hyperbolic direction
`hyper`	hyperbolic direction

Author(s)

Abdelaati Daouia and Thibault Laurent

References

Cazals et al. (2002), Nonparametric frontier estimation: a robust approach, Journal of Econometrics.

Daouia et al. (2009), Regularization of Nonparametric Frontier Estimators, TSE working paper.

Examples

# 1st example
data(spain)
score.orderm.b <- ordermscore.boot(xobs = as.matrix(spain[, c(2, 3, 4)]),
                    yobs = as.matrix(spain[, 1]))

system.time(
  ordermscore.boot(xobs = as.matrix(spain[, c(2, 3, 4)]), 
                 yobs = as.matrix(spain[, 1]))
)
system.time(
  ordermscore(xobs = as.matrix(spain[, c(2, 3, 4)]),
              yobs = as.matrix(spain[, 1]))
)

# 2nd example

data(burposte)
ind.samp <- sample(nrow(burposte), 500)
xobs <- as.matrix(burposte[ind.samp[1:100], 2])
yobs <- as.matrix(burposte[ind.samp[1:100], 3])
xeval <- as.matrix(burposte[ind.samp[101:500], 2])
yeval <- as.matrix(burposte[ind.samp[101:500], 3])
score.orderm.2.b <- ordermscore.boot(xobs, yobs, xeval, yeval)
 
# 1st example
data(spain)
score.orderm.b <- ordermscore.boot(xobs = as.matrix(spain[, c(2, 3, 4)]),
                    yobs = as.matrix(spain[, 1]))

system.time(
  ordermscore.boot(xobs = as.matrix(spain[, c(2, 3, 4)]), 
                 yobs = as.matrix(spain[, 1]))
)
system.time(
  ordermscore(xobs = as.matrix(spain[, c(2, 3, 4)]),
              yobs = as.matrix(spain[, 1]))
)

# 2nd example

data(burposte)
ind.samp <- sample(nrow(burposte), 500)
xobs <- as.matrix(burposte[ind.samp[1:100], 2])
yobs <- as.matrix(burposte[ind.samp[1:100], 3])
xeval <- as.matrix(burposte[ind.samp[101:500], 2])
yeval <- as.matrix(burposte[ind.samp[101:500], 3])
score.orderm.2.b <- ordermscore.boot(xobs, yobs, xeval, yeval)

The ROC curve to help choosing alpha and m parameters

Description

Computes the percentage of firms super-efficient according to the parameter alpha for alpha-quantile score and m for m-order score in a given direction.

Usage

ROCscore(xobs, yobs, type="output")
ROCscore(xobs, yobs, type="output")

Arguments

`xobs`	a matrix of size $n_1 \times p$ , input of sample points
`yobs`	a matrix of size $n_1 \times q$ , output of sample points
`type`	a direction to choose among "output", "input" and "hyper"

Details

A firm is super-efficient if it score is greater than 1. By consulting this graph, we may choose the values of alpha and m which correspond to the desired degree of robustness, i.e. the percentage of high performers of the population we want to exclude in our more realistic benchmarking comparison (see p.78 of Daraio and Simar, 2010).

Value

a data.frame object with:

`alpha`	different values of alpha
`f(alpha)`	the percentage of firms super-efficient
`m`	different values of m
`f(m)`	the percentage of firms super-efficient

Author(s)

Abdelaati Daouia and Thibault Laurent

References

Daraio and Simar (2007),Advanced Robust and Nonparametric Methods in Efficiency Analysis, Springer.

Examples

# 1st example
data(spain)
res.roc <- ROCscore(xobs = as.matrix(spain[,c(2,3,4)]),
                    yobs = as.matrix(spain[,1]),
                    type = "output")
# 1st example
data(spain)
res.roc <- ROCscore(xobs = as.matrix(spain[,c(2,3,4)]),
                    yobs = as.matrix(spain[,1]),
                    type = "output")

Spain data

Description

Spain data

Usage

data(spain)data(spain)

Format

A data frame with 61 observations on the following 4 variables.

y: a numeric vector of output
x1: a numeric vector of input
x2: a numeric vector of input
x3: a numeric vector of input

Examples

data(spain)
data(spain)

Package 'frontiles'

Help Index

alpha-quantile efficiency score and 2d/3d representation of frontier

Description

Details

Author(s)

References

Examples

Representation of alpha-quantile efficiency frontier for 1 output and 1 input

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

3d representation of alpha-quantile frontier in the case of 2 input and 1 output.

Description

Usage

Arguments

Details

Value

Note

Author(s)

See Also

Examples

Calculates alpha-quantile efficiency score

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Burposte data

Description

Usage

Format

References

Examples

Representation of m-order efficiency frontier for 1 output and 1 input

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Calculates order-m efficiency score with asymptotic formula

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Calculates order-m efficiency score with bootstrap algorithm

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

The ROC curve to help choosing alpha and m parameters

Description

Usage

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Author(s)