intF: 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

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

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

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

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

$$I_j(x) = \int_{x_j}^x \widehat{F}(r) d r = (x-x_j)\widehat{F}(x_j)+\Delta x_{j+1}\Bigl(\frac{\Delta x_{j+1}}{\Delta \widehat\phi_{j+1}}J\Bigl(\widehat\phi_j,\widehat\phi_{j+1}, \frac{x-x_j}{\Delta x_{j+1}}\Bigr)-\frac{\widehat f(x_j)(x-x_j)}{\Delta \widehat \phi_{j+1}}\Bigr)$$

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

Usage

intF(s, res)

Arguments

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 $\bold{s}$.

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.zb.unibe.ch/download/eldiss/06rufibach_k.pdf.

Examples

Run this code

# NOT RUN {
## 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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