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 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.
Vector of the same length as \(\bold{s}\), containing the values of \(\bar I\) at the elements of \(\bold{s}\).
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. Available at http://www.zb.unibe.ch/download/eldiss/06rufibach_k.pdf.
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).
# NOT RUN {
# for an example see the function intF.
# }
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