grenader {fdrtool}R Documentation

Grenander Estimator of a Decreasing Density

Description

The function grenander computes the Grenander estimator of a one-dimensional decreasing density.

Usage

grenander(F)

Arguments

F an ecdf containing the empirical cumulative density.

Details

The Grenander (1956) density estimator is given by the slopes of the least concave majorant (LCM) of the empirical distribution function (ECDF). It is a decreasing piecewise-constant function and can be shown to be the non-parametric maximum likelihood estimate (NPMLE) under the assumption of a decreasing density (note that the ECDF is the NPMLE without this assumption).

Value

A list of class grenander with the following components:

F the empirical distribution function specified as input.
x.knots x locations of the knots of the least concave majorant of the ECDF.
F.knots the corresponding y locations of the least concave majorant of the ECDF.
f.knots the corresponding slopes (=density).

Author(s)

Korbinian Strimmer (http://strimmerlab.org).

References

Grenander, U. (1956). On the theory of mortality measurement, Part II. Skan. Aktuarietidskr, 39, 125–153.

See Also

ecdf, gcmlcm, density.

Examples

# load "fdrtool" library
library("fdrtool")

# samples from random exponential variable 
z = rexp(30,1)
e = ecdf(z)
g = grenander(e)
g
plot(g) # plot ecdf, concave cdf, and Grenander estimator

# for comparison the kernel density estimate
plot(density(z)) 

# area under the Grenander density estimator 
sum( g$f.knots[-length(g$f.knots)]*diff(g$x.knots) )

[Package fdrtool version 1.1.1 Index]