| Title: | Statistical Functions for the Censored and Uncensored Epanechnikov Distribution |
|---|---|
| Description: | Analyzing censored variables usually requires the use of optimization algorithms. This package provides an alternative algebraic approach to the task of determining the expected value of a random censored variable with a known censoring point. Likewise this approach allows for the determination of the censoring point if the expected value is known. These results are derived under the assumption that the variable follows an Epanechnikov kernel distribution with known mean and range prior to censoring. Statistical functions related to the uncensored Epanechnikov distribution are also provided by this package. |
| Authors: | Mathias Borritz Milfeldt [aut, cre] |
| Maintainer: | Mathias Borritz Milfeldt <[email protected]> |
| License: | LGPL |
| Version: | 1.1.1 |
| Built: | 2024-12-08 07:04:52 UTC |
| Source: | CRAN |
This function calculates the censoring point of a random censored epanechnikov-distributed variable associated a given expected value.
The inverse of this function is evepan.
cepan(ev, mu = 0, r = 5^0.5, side_censored = "left")cepan(ev, mu = 0, r = 5^0.5, side_censored = "left")
ev |
expected value. |
mu |
mean of distribution prior to censoring. |
r |
half the range of the distribution, ie the distance from the mean to the smallest/largest value supported by the distribution. |
side_censored |
indicates whether the variable is |
the censoring point associated with ev, mu and r.
#Censoring point of a left-censored epan-distributed variable #with an expected value of 3 (given mu=0 and r=16): cepan(ev=3,mu=0,r=16) #Censoring point of a right-censored epan-distributed variable #with an expected value of 103 (given mu=100 and r=32): cepan(ev=94,mu=100,r=32,side_censored="right") #Results are usually not an integer though and rarely coinciding with mu#Censoring point of a left-censored epan-distributed variable #with an expected value of 3 (given mu=0 and r=16): cepan(ev=3,mu=0,r=16) #Censoring point of a right-censored epan-distributed variable #with an expected value of 103 (given mu=100 and r=32): cepan(ev=94,mu=100,r=32,side_censored="right") #Results are usually not an integer though and rarely coinciding with mu
This function is simply a polynomial of second degree.
depan(x = 0, mu = 0, r = 5^0.5)depan(x = 0, mu = 0, r = 5^0.5)
x |
point on x-axis. |
mu |
mean of distribution. |
r |
half the range of the distribution, ie the distance from the mean to the smallest/largest value supported by the distribution. |
point density associated with x, mu and r.
#Probability distribution function, epanechnikov: curve(depan(x),col='blue',ylim=c(0,.4),xlim=c(-3.5,3.5),yaxs='i',xaxs='i', main='Probability distribution function',ylab='Probability') #Probability distribution function, normal: curve(dnorm(x),col='green',add=TRUE) #Legend legend(x=-3.5,y=.4,legend=c('Epanechnikov pdf','Normal pdf'),lty=c(1,1),col=c('blue','green'))#Probability distribution function, epanechnikov: curve(depan(x),col='blue',ylim=c(0,.4),xlim=c(-3.5,3.5),yaxs='i',xaxs='i', main='Probability distribution function',ylab='Probability') #Probability distribution function, normal: curve(dnorm(x),col='green',add=TRUE) #Legend legend(x=-3.5,y=.4,legend=c('Epanechnikov pdf','Normal pdf'),lty=c(1,1),col=c('blue','green'))
This function calculates the expected value of a random censored epanechnikov-distributed variable with a given censoring point.
The inverse of this function is cepan.
evepan(c = 0, mu = 0, r = 5^0.5, side_censored = "left")evepan(c = 0, mu = 0, r = 5^0.5, side_censored = "left")
c |
censoring point. |
mu |
mean of distribution prior to censoring. |
r |
half the range of the distribution, ie the distance from the mean to the smallest/largest value supported by the distribution. |
side_censored |
indicates whether the variable is |
the expected value associated with c, mu and r.
#Expected value of an epan-distributed variable left-censored at 100 (given mu=100 and r=10): evepan(c=100,mu=100,r=10) #Expected value as a function of censoring point, epanechnikov distribution: curve(evepan(c=x),col='blue',xlim=c(-sqrt(5),sqrt(5)),yaxs='i',xaxs='i', main='Expected value as a function of censoring point',xlab='Censoring point',ylab='Expected value') #Expected value as a function of censoring point, normal distribution: curve(dnorm(x)+pnorm(x)*x,col='green',add=TRUE) #Expected value as a function of censoring point, no uncertainty: curve(1*x,col='grey',add=TRUE) #Legend legend(x=-sqrt(5),y=sqrt(5),legend=c('Epanechnikov','Normal distribution','No uncertainty'), lty=c(1,1),col=c('blue','green','grey'))#Expected value of an epan-distributed variable left-censored at 100 (given mu=100 and r=10): evepan(c=100,mu=100,r=10) #Expected value as a function of censoring point, epanechnikov distribution: curve(evepan(c=x),col='blue',xlim=c(-sqrt(5),sqrt(5)),yaxs='i',xaxs='i', main='Expected value as a function of censoring point',xlab='Censoring point',ylab='Expected value') #Expected value as a function of censoring point, normal distribution: curve(dnorm(x)+pnorm(x)*x,col='green',add=TRUE) #Expected value as a function of censoring point, no uncertainty: curve(1*x,col='grey',add=TRUE) #Legend legend(x=-sqrt(5),y=sqrt(5),legend=c('Epanechnikov','Normal distribution','No uncertainty'), lty=c(1,1),col=c('blue','green','grey'))
The inverse of this function is qepan.
pepan(x = 0, mu = 0, r = 5^0.5)pepan(x = 0, mu = 0, r = 5^0.5)
x |
point on x-axis. |
mu |
mean of distribution. |
r |
half the range of the distribution, ie the distance from the mean to the smallest/largest value supported by the distribution. |
probability of value below x given mu and r.
#Probability of a value below -1.96: pepan(x=-1.96,mu=0,r=5^.5) #Cumulative distribution function of epanechnikov distribution: curve(pepan(x),col='blue',xlim=c(-2.5,2.5),yaxs='i',xaxs='i', main='Cumulative distribution function',ylab='Probability') #Cumulative distribution function of standard normal distribution: curve(pnorm(x),col='green',add=TRUE) #Legend legend(x=-2.5,y=1,legend=c('Epanechnikov cdf','Normal cdf'),lty=c(1,1),col=c('blue','green'))#Probability of a value below -1.96: pepan(x=-1.96,mu=0,r=5^.5) #Cumulative distribution function of epanechnikov distribution: curve(pepan(x),col='blue',xlim=c(-2.5,2.5),yaxs='i',xaxs='i', main='Cumulative distribution function',ylab='Probability') #Cumulative distribution function of standard normal distribution: curve(pnorm(x),col='green',add=TRUE) #Legend legend(x=-2.5,y=1,legend=c('Epanechnikov cdf','Normal cdf'),lty=c(1,1),col=c('blue','green'))
The inverse of this function is pepan.
qepan(p, mu = 0, r = 5^0.5)qepan(p, mu = 0, r = 5^0.5)
p |
probability. |
mu |
mean of distribution. |
r |
half the range of the distribution, ie the distance from the mean to the smallest/largest value supported by the distribution. |
the quantile associated with x, mu and r.
#Calculate the lower quartile of an epan-distributed variable: qepan(p=.25,mu=0,r=sqrt(5)) #Use qepan to confirm analytical solution #Find the quantile corresponding to p=(5+sqrt(5))/8=.9045 when mu=0 and r=sqrt(5): qepan(p=(5+sqrt(5))/8,mu=0,r=sqrt(5)) #This is equal to (5-sqrt(5))/2#Calculate the lower quartile of an epan-distributed variable: qepan(p=.25,mu=0,r=sqrt(5)) #Use qepan to confirm analytical solution #Find the quantile corresponding to p=(5+sqrt(5))/8=.9045 when mu=0 and r=sqrt(5): qepan(p=(5+sqrt(5))/8,mu=0,r=sqrt(5)) #This is equal to (5-sqrt(5))/2
This function works in conjuncture with qepan and runif
repan(n, mu = 0, r = 5^0.5)repan(n, mu = 0, r = 5^0.5)
n |
number of data points. |
mu |
mean of distribution. |
r |
half the range of the distribution, ie the distance from the mean to the smallest/largest value supported by the distribution. |
vector of random variables.
#Generate and plot 10000 random observations: hist(repan(10000,mu=100,r=10))#Generate and plot 10000 random observations: hist(repan(10000,mu=100,r=10))