Calculate censoring point

Description

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.

Usage

cepan(ev, mu = 0, r = 5^0.5, side_censored = "left")

Arguments

Value

the censoring point associated with ev, mu and r.

Examples

#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

Probability density function (pdf) for an uncensored epanechnikov distribution

Description

This function is simply a polynomial of second degree.

Usage

depan(x = 0, mu = 0, r = 5^0.5)

Arguments

Value

point density associated with x, mu and r.

Examples

#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'))

Calculate expected value of censored variable

Description

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.

Usage

evepan(c = 0, mu = 0, r = 5^0.5, side_censored = "left")

Arguments

Value

the expected value associated with c, mu and r.

Examples

#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'))

Cumulative distribution function (cdf) for an uncensored epanechnikov distribution

Description

The inverse of this function is qepan.

Usage

pepan(x = 0, mu = 0, r = 5^0.5)

Arguments

Value

probability of value below x given mu and r.

Examples

#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'))

Quantile function for an uncensored epanechnikov distribution

Description

The inverse of this function is pepan.

Usage

qepan(p, mu = 0, r = 5^0.5)

Arguments

Value

the quantile associated with x, mu and r.

Examples

#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

Generate random uncensored epanechnikov-distributed data

Description

This function works in conjuncture with qepan and runif

Usage

repan(n, mu = 0, r = 5^0.5)

Arguments

Value

vector of random variables.

Examples

#Generate and plot 10000 random observations:
hist(repan(10000,mu=100,r=10))