intF: Computes the Integral of the estimated CDF at Arbitrary Real...

Description Usage Arguments Value Author(s) References See Also Examples

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

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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

Lutz Duembgen, duembgen@stat.unibe.ch,
http://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. 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.zb.unibe.ch/download/eldiss/06rufibach_k.pdf.

See Also

This function uses the output of activeSetLogCon. The function intECDF is similar, but based on the empirical distribution function.

Examples

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## 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)

logcondens documentation built on May 2, 2019, 6:11 a.m.