grenader {fdrtool} | R Documentation |
The function grenander
computes the Grenander estimator
of a one-dimensional decreasing density.
grenander(F)
F |
an ecdf containing the empirical cumulative density. |
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).
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://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(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) )