Computes the value of
$$\bar{I}(t) = \int_{x_1}^t \bar{F}(r) d \, r$$
where \(\bar F\) is the empirical distribution function of \(x_1,\ldots,x_m\), at all real numbers \(t\) in the vector \(\bold{s}\). Note that \(t\) (so all elements in \(\bold{s}\)) must lie in \([x_1,x_m]\). The exact formula for \(\bar I(t)\) is
$$\bar I(t) = \Big(\sum_{i=2}^{i_0}(x_i-x_{i-1})\frac{i-1}{n} \Big) + (t-x_{i_0})\frac{i_0-1}{n}$$
where \(i_0 = \max_{i=1,\ldots,m} \{x_i \le t\}\).
intECDF(s, x)Vector of the same length as \(\bold{s}\), containing the values of \(\bar I\) at the elements of \(\bold{s}\).
Vector of real numbers in \([x_1,x_m]\) where \(\bar{I}\) should be evaluated at.
Vector \({\bold{x}} = (x_1, \ldots, x_m)\) of original observations.
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
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. tools:::Rd_expr_doi("https://doi.org/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.
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).
# for an example see the function intF.
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