Grenander Estimator of a Decreasing or Increasing Density
The function grenander computes the Grenander estimator
of a one-dimensional decreasing or increasing density.
grenander(F, type=c("decreasing", "increasing"))F |
an |
type |
specifies whether the distribution is decreasing (the default) or increasing. |
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). Similarly, an increasing density function is obtained by using the greatest convex minorant (GCM) of the ECDF.
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). |
Korbinian Strimmer (http://www.strimmerlab.org).
Grenander, U. (1956). On the theory of mortality measurement, Part II. Skan. Aktuarietidskr, 39, 125–153.
# load "fdrtool" library
library("fdrtool")
# samples from random exponential variable
z = rexp(30,1)
e = ecdf(z)
g = grenander(e)
g
# plot ecdf, concave cdf, and Grenander density estimator (on log scale)
plot(g, log="y")
# 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) )Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.