Description Usage Arguments Details Value Author(s) References Examples

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

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

`dnum` |
number of distribution. |

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]*.

`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 |

`breaks` |
gives the vector of break points. |

Thoralf Mildenberger

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

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