Computes the Integral of the estimated CDF at Arbitrary Real Numbers in s
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.
intF(s, res)
s |
Vector of real numbers where the functions should be evaluated at. |
res |
An object of class |
Vector of the same length as \bold{s}, containing the values of \widehat I at the elements of s.
Kaspar Rufibach, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch
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.
This function uses the output of activeSetLogCon. The function intECDF is similar, but based
on the empirical distribution function.
## 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)
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