Package {DENSaftertransform}

Adaptive log-normal plug-in density estimator for power transformations.

Description

Computing the adaptive log-normal PI kernel density estimate for nonnegative data after a power transformation with exponent ranging from 0 to 1 based on the Gaussian kernel.

Usage

KDE_PI_adapt_lognorm_power(x, alpha, u)

Arguments

Value

Vector of density estimates at the corresponding values of the argument (u).

References

Savchuk, O. (2026, submitted). Adaptive log-normal plug-in density estimator for power transformations.

See Also

KDE_power_PI.

Examples

#Example (standard exponential density, n=1000, alpha=1/5)
# Plotting the true density g of Y = X^alpha, where X ~ Exp(1) (black)
# together with the adaptive PI estimate(blue) and the
# single-bandwidth PI estimate(olive).
n=1000
alpha=1/5
u=seq(0.001,2,len=1000) #estimation points
x_exp=rexp(n)
h_PI_exp=h_pi_datadriven_power(x_exp,alpha)
#true density after the power transformation of the standard exponential distribution
g_exp=u^(1/alpha-1)*exp(-u^(1/alpha))/alpha  #true density after the power transformation
KDE_exp_adapt_PI=KDE_PI_adapt_lognorm_power(x_exp,alpha,u)
KDE_exp_PI=KDE_power_PI(alpha,h_PI_exp,x_exp,u)
dev.new()
plot(u,g_exp,'l',lwd=2,col="black",xlab="y",ylab="density",cex.lab=1.7,cex.axis=1.7,cex.main=1.7,
main="Standard exponential distribution",ylim=c(0,1.25*max(KDE_exp_adapt_PI)))
lines(u,KDE_exp_adapt_PI,lty="solid",lwd=2,col="blue")
lines(u,KDE_exp_PI,lty="dashed",lwd=2,col="forestgreen")
legend("topleft", legend = c("Adaptive PI estimate", "PI estimate", "density g"),
lty=c("solid","dashed","solid"), col=c("blue","forestgreen","black"),lwd=2,bty="n",cex=1.7)
legend("topright",legend=expression(n==1000,alpha==frac(1,5)),cex=1.7,bty="n",y.intersp=1.3)

Adaptive normal PI estimate in the case of the logarithmic transformation.

Description

Computing the adaptive normal PI kernel density estimate for the log-transformed data based on the Gaussian kernel.

Usage

KDE_PI_adapt_norm(x, u)

Arguments

Value

Vector of density estimates at the corresponding values of the argument (u).

References

Savchuk, O. (2026). Density estimation for log-transformed data. Computational Statistics, 41(98).

See Also

KDE_ln_PI.

Examples

#Example(skewed bimodal density). Estimating the density corresponding to the logarithmic 
#transformation of a sample of size n=1000 originated from the skewed bimodal density 
#(density #8 of Marron and Wand (1992)).
#Density after the logarithmic transformation
g_skew=function(u)
  3/4*dnorm(u)+1/4*dnorm(u,mean=3/2,sd=1/3) 
skew_realiz=function(n){
  # n is the sample size
  mix_comp=runif(n)<0.25
  dat=numeric(n)
  dat[mix_comp]=rnorm(sum(mix_comp),mean=3/2,sd=1/3)
  dat[!mix_comp]=rnorm(sum(!mix_comp))
 dat
}
#Plotting an estimate
n=1000
u=seq(-4,4,len=1000)
y_skew=skew_realiz(n)
x_skew=exp(y_skew)
h_PI=h_ln_PI(x_skew)
KDE_skew_PI=KDE_ln_PI(h_PI,x_skew,u)
KDE_skew_adapt_norm=KDE_PI_adapt_norm(x_skew,u)
dev.new()
plot(u,g_skew(u),'l',lwd=2,ylim=c(0,1.05*max(KDE_skew_PI)),col="red",xlab="y",ylab="density",
cex.lab=1.7,cex.axis=1.7,cex.main=1.7,main="Skewed bimodal distribution")
lines(u,KDE_skew_PI,lty="longdash",lwd=2)
lines(u,KDE_skew_adapt_norm,lty="solid",lwd=2,col="blue")
legend(-4.5,0.45,legend=c("PI estimate","Adaptive normal","density g"),lty=c("longdash",
"solid","solid"),col=c("black","blue","red"),lwd=c(3,3,3),bty="n",cex=1.7)
legend(1.25,0.45,legend=paste("n=",n),cex=2,bty="n")

The PI kernel density estimate in the case of the logarithmic transformation.

Description

Computing the PI kernel density estimate for the log-transformed data based on the Gaussian kernel.

Usage

KDE_ln_PI(h, x, u)

Arguments

Value

Vector of density estimates at the corresponding values of the argument (u).

References

Savchuk, O. (2026). Density estimation for log-transformed data. Computational Statistics, 41(98).

See Also

h_ln_PI.

Examples

#Example(Gamma(shape=2,scale=2) density). Estimating the pdf of the density corresponding to the
#logarithmic transformation of sample of size n=100 originated from the gamma density.
xdat=rgamma(100,2,scale=2)  #original data
n=length(xdat)  #sample size
u=seq(-3,5,len=1000) #estimation points
dens_lntransform=exp(2*u-exp(u)/2)/4  #density after log-transformation
h_PI=h_ln_PI(xdat)
dev.new()
dens_PI=KDE_ln_PI(h_PI,xdat,u)
plot(u,dens_PI,'l',ylim=c(0,1.05*max(dens_PI)),lwd=2,cex.lab=1.7,cex.axis=1.7,cex.main=1.5,
main="Gamma dens. after ln-transform. and its PI estimate",xlab="u",ylab="density")
lines(u,dens_lntransform,lwd=2,lty="dashed",col="red",bty="n")
legend(-3,0.5,legend=c("true density","PI estimate"),lwd=c(2,2),col=c("red","black"),
lty=c("dashed","solid"),cex=1.5,bty="n")
text(3.25,0.525,paste("n=",n),cex=2)
text(3.85,0.425,bquote(h[PI] == .(sprintf("%.4f", h_PI))),cex=2)

The PI kernel density estimate in the case of the power transformation.

Description

Computing the PI kernel density estimate for data after a power transformation based on the Gaussian kernel.

Usage

KDE_power_PI(alpha, h, x, u)

Arguments

Value

Vector of density estimates at the corresponding values of the argument (u).

References

1. Savchuk, O. and Schick, A. (2013). Density estimation for power transformations. Journal of Nonparametric Statistics, 25(3), 545–559. doi:10.1080/10485252.2013.811788

2. Savchuk, O. (2026, submitted). Adaptive log-normal plug-in density estimator for power transformations.

See Also

KDE_ln_PI, h_pi_datadriven_power.

Examples

#Example(original data is standard normal, n=1000 and (1)alpha=0.5 (Case 1), (2) alpha=2 (Case 2))
n=1000
xdat=rnorm(n)
h_P=bw.SJ(xdat) #Sheather-Jones PI for original x-data

#Case1: alpha=0.5
alpha=0.5
u=seq(-2.5,2.5,len=1000)
g_0.5=1/alpha/sqrt(2*pi)*abs(u)^(1/alpha-1)*exp(-abs(u)^(2/alpha)/2) #true density
PI_power_est_0.5=KDE_power_PI(alpha,h_P,xdat,u)
dev.new()
plot(u,g_0.5,lwd=2,'l',col="red",ylim=c(0,1.125*max(PI_power_est_0.5)),xlab="y",ylab="density",
cex.axis=1.7,cex.main=1.7,cex.lab=1.7,main=expression(paste("Norm density, ",alpha,"=0.5")))
lines(u,PI_power_est_0.5,lwd=2,lty="dashed")
legend(-2.75,1.2*max(max(PI_power_est_0.5),0.52),legend=c("true density","PI estimate"),
lty=c("solid","dashed"),col=c("red","black"),lwd=c(2,2),bty="n",cex=1.5)
legend(0.75,0.5,legend=paste("n=",n),cex=2,bty="n")

#Case 2: alpha=2
alpha=2
y=sign(xdat)*abs(xdat)^alpha
u=seq(-1.5,1.5,len=1000)
g_2=1/alpha/sqrt(2*pi)*abs(u)^(1/alpha-1)*exp(-abs(u)^(2/alpha)/2) #true density
PI_power_est_2=KDE_power_PI(alpha,h_P,xdat,u)
dev.new()
plot(u,g_2,lwd=2,'l',col="red",ylim=c(0,1.1*max(PI_power_est_2)),xlab="y",ylab="density",
cex.axis=1.7,cex.main=1.7,cex.lab=1.7,main=expression(paste("Norm density, ",alpha,"=2")))
lines(u,PI_power_est_2,lwd=2,lty="dashed")
legend(0,2,legend=c("true density","PI estimate"),lty=c("solid","dashed"),col=c("red","black"),
lwd=c(2,2),cex=1.5,bty="n")
legend(-1.5,5,legend=paste("n=",n),cex=2,bty="n")

The bandwidth for the single-bandwidth PI density estimator in the case of the logarithmic transformation.

Description

Bandwidth for the PI estimator for estimating g, the true pdf of the transformed sample.

Usage

h_ln_PI(x)

Arguments

Value

The bandwidth for the PI estimator for estimating g, the density after the logarithmic transformation.

References

Savchuk, O. (2026). Density estimation for log-transformed data. Computational Statistics, 41(98).

See Also

KDE_ln_PI.

Examples

# Example. Finding bandwidth for the PI estimator for the log-transformed sample of size n=100 
#originated from a gamma distribution 
#with the parameters: shape=2, scale=2.
xdat=rgamma(100,2,scale=2)
paste("PI bandwidth for the PI estimator=", h_ln_PI(xdat))

The bandwidth for the single-bandwidth PI density estimator for data after the power transformation with an exponent ranging from zero to one based on the Gaussian kernel.

Description

Bandwidth for the PI estimator for estimating g, the true pdf of the transformed sample.

Usage

h_pi_datadriven_power(x, alpha)

Arguments

Value

The bandwidth for the PI density estimator for estimating g, the density after the power transformation.

References

Savchuk, O. (2026, submitted). Adaptive log-normal plug-in density estimator for power transformations.

See Also

KDE_power_PI, KDE_PI_adapt_lognorm_power.

Examples

# Example. Finding bandwidth for the single-bandwidth PI density estimator for the transformed 
# data on river lengths of 141 major North American rivers with alpha=1/5.
xdat=rivers
alpha=1/5
n=length(xdat)
h_PI_power=h_pi_datadriven_power(xdat,alpha)
u=seq(2,5.5,len=1000) #estimation points
dens_PI=KDE_power_PI(alpha,h_PI_power,xdat,u)
dens_TD=density(xdat^alpha,bw="ucv",from=2,to=5.5,n=1000)
dev.new()
oldpar=par(mar=c(6.5, 5, 2, 2))
plot(u,dens_PI,'l',lwd=2,xlab=expression(length^{frac(1,5)}),ylab="density",cex.lab=1.7,
cex.axis=1.7)
lines(dens_TD,lwd=2,col="red")
legend("topright",legend=as.expression(c(bquote(n==.(n)),bquote(alpha==frac(1,5)),
bquote(h[PI]==.(round(h_PI_power,2))),bquote(h[CV]==.(round(dens_TD$bw,2))))),
col=c(NA,NA,"black","red"),lwd=c(NA,NA,2,2),cex=1.7,bty="n",y.intersp=1.2)
legend("topleft",legend=c("PI","TD"),lwd=c(2,2),col=c("black","red"),bty="n",cex=1.7)
title("Density of river lengths",cex=1.5)
par(oldpar)