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

Based on the output of the function `activeSetLogCon`

, this 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(∑_{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 ≤ t \} \} * and

*I_j(x) = \int_{x_j}^x \widehat{F}(r) d r =
(x-x_j)\widehat{F}(x_j)+Δ x_{j+1}\Bigl(\frac{Δ x_{j+1}}{Δ
\widehat\varphi_{j+1}}J\Bigl(\widehat\varphi_j,\widehat\varphi_{j+1},
\frac{x-x_j}{Δ x_{j+1}}\Bigr)-\frac{\widehat f(x_j)(x-x_j)}{Δ
\widehat \varphi_{j+1}}\Bigr)*

for *x \in [x_j, x_{j+1}], \ j = 1,…,m-1*, *Δ v_{i+1}
= v_{i+1}-v_i* for any vector *\bold{v}* and the function *J*
introduced in `Jfunctions`

.

Note that this version of intF is similar to that in the logcondens package, versions 1.3.5 and earlier. Newer versions of that package have modified arguments. Here, we have also added the argument 'prec'.

1 | ```
intF(s, x, phi, Fhat, prec=1e-10)
``` |

`s` |
Vector of real numbers where the functions should be evaluated at. |

`x` |
Vector |

`phi` |
Vector |

`Fhat` |
Vector
as computed by |

`prec` |
Governs cutoff at which an approximation for the needed integral is used. |

Vector of the same length as *\bold{s}*, containing the values of *\widehat I* at the elements of *\bold{s}*.

Kaspar Rufibach, kaspar.rufibach@ifspm.uzh.ch,

http://www.biostat.uzh.ch/aboutus/people/rufibach.html

Lutz Duembgen, duembgen@stat.unibe.ch,

http://www.staff.unibe.ch/duembgen

Duembgen, L, Huesler, A. and Rufibach, K. (2010) Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at http://arxiv.org/abs/0707.4643.

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

Doss, C. R. (2013). Shape-Constrained Inference for Concave-Transformed Densities and their Modes. PhD thesis, Department of Statistics, University of Washington, in preparation.

Doss, C. R. and Wellner, J. A. (2013). Inference for the mode of a log-concave density. Technical Report, University of Washington, in preparation.

This function uses the output of
`activeSetLogCon`

. The function `intECDF`

is
similar, but based on the empirical distribution function.

1 2 3 4 5 6 7 8 9 10 11 12 | ```
## estimate gamma density
set.seed(1977)
x <- sort(rgamma(200, 2, 1))
res <- activeSetLogCon(x, w = NA, print = FALSE)
## compute and plot the process D(t) in Duembgen and Rufibach (2009)
s <- seq(min(x), max(x), by = 10 ^ -3)
D1 <- intF(s, x, res$phi, res$Fhat)
D2 <- intECDF(s, x)
par(mfrow = c(2, 1))
plot(x, res$phi, type = 'l'); rug(x)
plot(s, D1 - D2, type = 'l'); abline(h = 0, lty = 2)
``` |

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.