# histo: Some properties of 4 histogram benchmark densities In benchden: 28 benchmark densities from Berlinet/Devroye (1994)

## Description

Name, position of modes, support and break points for the 4 histogram benchmark distributions from Rozenholc/Mildenberger/Gather (2010).

## Usage

 `1` ```histo(dnum = 1) ```

## Arguments

 `dnum` number of distribution.

## Details

These functions implement the 4 histogram benchmark distributions from Rozenholc/Mildenberger/Gather (2010). Defined as the following mixtures of uniform distributions:

`dnum == 1` 5 bin regular histogram:

0.15*U[0,0.2] + 0.35*U(0.2,0.4] + 0.2*U(0.4,0.6] +0.1*U(0.6,0.8]+ 0.2*U(0.8,1.0]

`dnum == 2` 5 bin irregular histogram:

0.15*U[0,0.13] + 0,35*U(0.13,0.34] + 0.2*U(0.34,0.61] +0.1*U(0.61,0.65] + 0.2*U(0.65,1.0]

`dnum == 3` 10 bin regular histogram:

0.01*U[0,0.1] + 0.18*U(0.1,0.2] + 0.16*U(0.2,0.3]

+0.07*U(0.3,0.4] + 0.06*U(0.4,0.5] + 0.01*U(0.5,0.6]

+0.06*U(0.6,0.7] + 0.37*U(0.7,0.8] + 0.06*U(0.8,0.9]

+0.02*U(0.9,1.0]

`dnum == 4` 10 bin irregular histogram:

0.01*U[0,0.02] + 0.18*U(0.02,0.07] + 0.16*U(0.07,0.14]

+0.07*U(0.14,0.44] + 0.06*U(0.44,0.53] + 0.01*U(0.53,0.56]

+0.06*U(0.56,0.67] + 0.37*U(0.67,0.77] + 0.06*U(0.77,0.91]

+0.02*U(0.91,1.0]

where U[a,b] denotes the uniform distribution on [a,b].

## Value

`histo` returns a list with the following components:

 `name` gives the name of the distribution. `peaks` gives a vector of the positions of peaks of the density, defined here as mid points of maximal intervals. `support` gives a matrix with one row with the endpoints of the support, which is [0,1] for all four histogram densities. `breaks` gives the vector of break points.

## Author(s)

Thoralf Mildenberger

## References

T. Mildenberger and H. Weinert, "The benchden Package: Benchmark Densities for Nonparametric Density Estimation", Journal of Statistical Software, vol. 46(14), 1-14, 2012. http://www.jstatsoft.org/v46/i14/

Y. Rozenholc, T. Mildenberger and U. Gather (2010), "Combining Regular and Irregular Histograms by Penalized Likelihood", Computational Statistics and Data Analysis, 54, 3313-3323. Earlier version including explicit definition of the densities: http://www.statistik.tu-dortmund.de/fileadmin/user_upload/SFB_823/discussion_papers/2009/31_09_rozenholc_mildenberger_gather.pdf

## Examples

 ```1 2 3 4 5``` ```# position of peaks of the 5 bin irregular histogram density histo(dnum=2)\$peaks # support of the 10 bin regular histogram density histo(dnum=3)\$support ```

benchden documentation built on May 2, 2019, 3:40 p.m.