dhisto {benchden}

4 histogram benchmark densities

Description

Density, distribution function, quantile function and random variate generation for the 4 histogram benchmark distributions from Rozenholc/Mildenberger/Gather (2010).

Usage

dhisto(x,dnum = 1)
phisto(q,dnum = 1)
qhisto(p,dnum = 1)
rhisto(n,dnum = 1)

Arguments

dnum

number of distribution as in Rozenholc/Mildenberger/Gather (2010)

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations.

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

dhisto

gives the density,

phisto

gives the distribution function,

qhisto

gives the quantile function, and

rhisto

generates random deviates.

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. 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. doi:10.1016/j.csda.2010.04.021 Earlier version including explicit definition of the densities: doi:10.17877/DE290R-15901

Examples

# histogram and true density of "5 bin irregular"-distribution
hist(rhisto(2000,dnum=2),breaks=250, main = " ",freq=FALSE)
lines(seq(0,1,0.01),dhisto(seq(0,1,0.01),dnum=2),col="blue",lwd=1)
title(paste("sample from",nhisto(dnum=2),"density"))