intECDFfn: Gives the Integrated Empirical Distribution Function

Description

Like intECDF, except returns a function $\bar I$ instead of a value. The function $\bar I(l,r)$ is given by

$$\bar{I}(l,r) = \int_{l}^{r} \bar{F}(u) d \, u$$

where $\bar F$ is the empirical distribution function of $x_1,\ldots,x_m$. Note that $l$ and $r$ must lie in $[x_1,x_m]$.

For an exact formula related to $\bar{I}$, see intECDF.

Usage

intECDFfn(x)

Arguments

Vector $x = (x_1, \ldots, x_m)$ of original observations, which are used to define the empirical CDF, $\bar{F}$.

Value

The function $\bar I$.

References

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.

Examples

Run this code


set.seed(100)
xx <- runif(50) ## min .056, max .88

myIntECDF <- intECDFfn(xx);
evalpts <- c(.3, .5) ## lie within [ min(xx) , max(xx) ]
myIntECDF(evalpts)
## equal to
intECDF(evalpts, xx)