Package {DENSaftertransform}

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, under review). Density estimation for log-transformed data.

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 logarithmi 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 logaritjmic 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, under review). Density estimation for log-transformed data.

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

Savchuk, O., Schick A. (2013). Density estimation for power transformations. Journal of Nonparametric Statistics, 25(3), 545-559.

See Also

KDE_ln_PI.

Examples

#Example(original data is sandard 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")

Bandwidth for the PI estimator in the case of the logarithmic transformation.

Description

Bandwidth for the PI estimator for estimating g, the true probability density function (pdf), correponding to the logarithmic transformation of data based on replacing g by the reference N(\bar y,s_Y^2) distribution. See Section 7 of the article by Savchuk (2026).

Usage

h_ln_PI(x)

Arguments

Value

The bandwidth for the PI estimator for estimating g, the density after the logarithmic transformation, obtained by replacing g by the reference N(\bar y,s_Y^2) density.

References

Savchuk, O. (2026, under review). Density estimation for log-transformed data.

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))