intECDF {logcondens}R Documentation

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,...,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) = 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,...,m} {x_i <= 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 {x} = (x_1, ..., x_m) of original observations (sorted).

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

Lutz Duembgen, duembgen@stat.unibe.ch,
http://www.staff.unibe.ch/duembgen

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.

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.stub.unibe.ch/download/eldiss/06rufibach_k.pdf.

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.

[Package logcondens version 1.3.3 Index]