logcondens (version 2.0.6)

intECDF: 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 $\bold{s}$.

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. 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.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.