intF {logcondens}R Documentation

Computes the Integral of the estimated CDF at Arbitrary Real Numbers in s

Description

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

I^(t)=x1tF^(r)dr\widehat I(t) = \int_{x_1}^t \widehat{F}(r) d r

at all numbers in s\bold{s}. Note that tt (so all elements in s\bold{s}) must lie in [x1,xm][x_1,x_m]. The exact formula for I^(t)\widehat I(t) is

I^(t)=(i=1i0I^i(xi+1))+I^i0(t)\widehat I(t) = \Bigl(\sum_{i=1}^{i_0} \widehat{I}_i(x_{i+1})\Bigr)+\widehat{I}_{i_0}(t)

where i0=i_0 = min{m1, {i : xit}}\{m-1 \, , \ \{i \ : \ x_i \le t \}\} and

Ij(x)=xjxF^(r)dr=(xxj)F^(xj)+Δxj+1(Δxj+1Δϕ^j+1J(ϕ^j,ϕ^j+1,xxjΔxj+1)f^(xj)(xxj)Δϕ^j+1)I_j(x) = \int_{x_j}^x \widehat{F}(r) d r = (x-x_j)\widehat{F}(x_j)+\Delta x_{j+1}\Bigl(\frac{\Delta x_{j+1}}{\Delta \widehat\phi_{j+1}}J\Bigl(\widehat\phi_j,\widehat\phi_{j+1}, \frac{x-x_j}{\Delta x_{j+1}}\Bigr)-\frac{\widehat f(x_j)(x-x_j)}{\Delta \widehat \phi_{j+1}}\Bigr)

for x[xj,xj+1], j=1,,m1x \in [x_j, x_{j+1}], \ j = 1,\ldots, m-1, Δvi+1=vi+1vi\Delta v_{i+1} = v_{i+1} - v_i for any vector v\bold{v} and the function JJ introduced in Jfunctions.

Usage

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 s\bold{s}, containing the values of I^\widehat I at the elements of s\bold{s}.

Author(s)

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

Lutz Duembgen, duembgen@stat.unibe.ch,
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 uses the output of activeSetLogCon. The function intECDF is similar, but based on the empirical distribution function.

Examples

## 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)

[Package logcondens version 2.1.8 Index]