benchden (version 1.0.8)

histo: Some properties of 4 histogram benchmark densities

Description

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

Usage

histo(dnum = 1)

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.

Arguments

dnum

number of distribution.

Author

Thoralf Mildenberger

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 \([a,b]\).

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. https://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. tools:::Rd_expr_doi("10.1016/j.csda.2010.04.021") Earlier version including explicit definition of the densities: tools:::Rd_expr_doi("10.17877/DE290R-15901")

Examples

# 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