# intECDF: Computes the Integrated Empirical Distribution Function at... In logcondens: Estimate a Log-Concave Probability Density from Iid Observations

## Description

Computes the value of

\bar{I}(t) = \int_{x_1}^t \bar{F}(r) dr

where \bar F is the empirical distribution function of x_1,…,x_m, at all real numbers t in the vector s. Note that t (so all elements in s) must lie in [x_1,x_m]. The exact formula for \bar I(t) is

\bar I(t) = (∑_{i=2}^{i_0}(x_i-x_{i-1}) (i-1)/n) + (t-x_{i_0})(i_0-1)/n

where i_0 = \max_{i=1,…, m}{x_i ≤ t}.

## Usage

 1 intECDF(s, x) 

## Arguments

 s Vector of real numbers in [x_1,x_m] where \bar{I} should be evaluated at. x Vector x = (x_1, …, x_m) of original observations.

## Value

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

## Author(s)

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

## 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. http://www.jstatsoft.org/v39/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.
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).
 1 # for an example see the function intF.