intECDF {logcondens}R Documentation

Computes the Integrated Empirical Distribution Function at Arbitrary Real Numbers in s

Description

Computes the value of

Iˉ(t)=x1tFˉ(r)dr\bar{I}(t) = \int_{x_1}^t \bar{F}(r) d \, r

where Fˉ\bar F is the empirical distribution function of x1,,xmx_1,\ldots,x_m, at all real numbers tt in the vector s\bold{s}. Note that tt (so all elements in s\bold{s}) must lie in [x1,xm][x_1,x_m]. The exact formula for Iˉ(t)\bar I(t) is

Iˉ(t)=(i=2i0(xixi1)i1n)+(txi0)i01n\bar I(t) = \Big(\sum_{i=2}^{i_0}(x_i-x_{i-1})\frac{i-1}{n} \Big) + (t-x_{i_0})\frac{i_0-1}{n}

where i0=maxi=1,,m{xit}i_0 = \max_{i=1,\ldots,m} \{x_i \le t\}.

Usage

intECDF(s, x)

Arguments

s

Vector of real numbers in [x1,xm][x_1,x_m] where Iˉ\bar{I} should be evaluated at.

x

Vector x=(x1,,xm){\bold{x}} = (x_1, \ldots, x_m) of original observations.

Value

Vector of the same length as s\bold{s}, containing the values of Iˉ\bar I at the elements of s\bold{s}.

Author(s)

Kaspar Rufibach, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch

Lutz Duembgen, duembgen@stat.unibe.ch,
https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html

References

Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log–concave density and its distribution function: basic properties and uniform consistency. Bernoulli, 15(1), 40–68.

Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. Journal of Statistical Software, 39(6), 1–28. doi:10.18637/jss.v039.i06

Rufibach K. (2006) Log-concave Density Estimation and Bump Hunting for i.i.d. Observations. PhD Thesis, University of Bern, Switzerland and Georg-August University of Goettingen, Germany, 2006.
Available at https://slsp-ube.primo.exlibrisgroup.com/permalink/41SLSP_UBE/17e6d97/alma99116730175505511.

See Also

This function together with intF can be used to check the characterization of the log-concave density estimator in terms of distribution functions, see Rufibach (2006) and Duembgen and Rufibach (2009).

Examples

# for an example see the function intF.

[Package logcondens version 2.1.8 Index]