eff.dens {Benchmarking} R Documentation

## Estimate and plot density of efficiencies

### Description

A method to estimate and plot kernel estimate of (Farrell) efficiencies taken into consideration that efficiencies are bounded either above (input direction) or below (output direction).

### Usage

```eff.dens(eff, bw = "nrd0")

eff.dens.plot(obj, bw = "nrd0", ..., xlim, ylim, xlab, ylab)
```

### Arguments

 `eff` Either a list of (Farrell) efficiencies or a Farrell object returned from the method `dea`. `bw` Bandwith, look at the documentation of `density` for an explanation. `obj` Either an array of efficiencies or a list returned from `eff.dens`. `...` Further arguments to the `plot` method like line type and line width. `xlim` Range on the x-axis; usualy not needed, just use the defaults. `ylim` Range on the x-axis; usualy not needed, just use the defaults. `xlab` Label for the x-axis. `ylab` Label for the y-axis.

### Details

The calculation is based on a reflection method (Silverman 1986, 30) using the default window kernel and defult bandwidth (window width) in the method `density`.

The method `eff.dens.plot` plot the density directly, and `eff.dens` just estimate the numerical density, and the result can then either be plotted by `plot`, corresponds to `eff.dens.plot`, or by lines as an overlay on an existing plot.

### Value

The return from `eff.dens` is a list `list(x,y)` with efficiencies and the corresponding density values.

### Note

The input efficiency is also bounded below by 0, but for normal firms an efficiency at 0 will not happen, i.e. the boundary is not effective, and therefore this boundary is not taken into consideration.

### Author(s)

Peter Bogetoft and Lars Otto larsot23@gmail.com

### References

B.W. Silverman (1986), Density Estimation for Statistics and Data Analysis, Chapman and Hall, London.

### Examples

```e <- 1 - rnorm(100)
e[e>1] <- 1
e <- e[e>0]
eff.dens.plot(e)

hist(e, breaks=15, freq=FALSE, xlab="Efficiency", main="")
den <- eff.dens(e)
lines(den,lw=2)
```

[Package Benchmarking version 0.29 Index]