Computes the Integrated Empirical Distribution Function at Arbitrary Real Numbers in s
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}.
intECDF(s, x)
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. |
Vector of the same length as s, containing the values of \bar I at the elements of s.
Kaspar Rufibach, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch
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).
# for an example see the function intF.
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