intECDF {logcondens}

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

Description

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\}.

Usage

intECDF(s, x)

Arguments

s

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

x

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

Value

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

Author(s)

Kaspar Rufibach, [e-mail],
http://www.kasparrufibach.ch

Lutz Duembgen, [e-mail],
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.