# intF: Computes the Integral of the estimated CDF at Arbitrary Real... In logcondens: Estimate a Log-Concave Probability Density from Iid Observations

## Description

Based on an object of class dlc as output by the function logConDens, this function gives values of

\hat I(t) = \int_{x_1}^t \hat{F}(r) d r

at all numbers in s. Note that t (so all elements in s) must lie in [x_1,x_m]. The exact formula for \widehat I(t) is

\hat I(t) = (∑_{i=1}^{i_0} \hat{I}_i(x_{i+1}))+\hat{I}_{i_0}(t)

where i_0 = min{m-1, {i : x_i ≤ t}} and

I_j(x) = int_{x_j}^x \hat{F}(r) d r = (x-x_j)\hat{F}(x_j)+Δ x_{j+1}((Δ x_{j+1})/(Δ \hatφ_{j+1})J(\hatφ_j, \hat φ_{j+1}, (x-x_j)/(Δ x_{j+1}))-(\hat f(x_j)(x-x_j))/(Δ \hat φ_{j+1}))

for x \in [x_j, x_{j+1}], j = 1, …, m-1, Δ v_{i+1} = v_{i+1} - v_i for any vector v and the function J introduced in Jfunctions.

## Usage

 1 intF(s, res) 

## Arguments

 s Vector of real numbers where the functions should be evaluated at. res An object of class "dlc", usually a result of a call to logConDens.

## Value

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

## Author(s)

Kaspar Rufibach, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch

## 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.
This function uses the output of activeSetLogCon. The function intECDF is similar, but based on the empirical distribution function.
  1 2 3 4 5 6 7 8 9 10 11 12 ## estimate gamma density set.seed(1977) x <- rgamma(200, 2, 1) res <- logConDens(x, smoothed = FALSE, print = FALSE) ## compute and plot the process D(t) in Duembgen and Rufibach (2009) s <- seq(min(res$x), max(res$x), by = 10 ^ -3) D1 <- intF(s, res) D2 <- intECDF(s, res$xn) par(mfrow = c(2, 1)) plot(res$x, res$phi, type = 'l'); rug(res$x) plot(s, D1 - D2, type = 'l'); abline(h = 0, lty = 2)