logcondens (version 2.1.8)

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

$$\hat{I}(t)={\int }_{x_1}^t\hat{F}(r)dr$$

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

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

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

$$I_j(x)={\int }_{x_j}^x\hat{F}(r)dr=(x-x_j)\hat{F}(x_j)+\mathrm{\Delta }{x}_{j+1}(\frac{\mathrm{\Delta }{x}_{j+1}}{\mathrm{\Delta }{\hat{\varphi }}_{j+1}}J({\hat{\varphi }}_j,{\hat{\varphi }}_{j+1},\frac{x-x_j}{\mathrm{\Delta }{x}_{j+1}})-\frac{\hat{f}(x_j)(x-x_j)}{\mathrm{\Delta }{\hat{\varphi }}_{j+1}})$$

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

Usage

intF(s, res)

Value

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

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.

Author

Kaspar Rufibach, ,
http://www.kasparrufibach.ch

Lutz Duembgen, ,
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. tools:::Rd_expr_doi("https://doi.org/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)