TNorm function

Truncated normal distribution

Random generation for the truncated normal distribution. The mean and standard deviation of the original normal distribution are mean and sd. Truncation limits are given by a, b, type of truncation is given by trunc.

rTNorm(n, mean=0, sd=1, a, b, trunc)

Arguments

• mean: mean (if common for all observations) or a vector of length n of means.

• sd: standard deviation (if common for all observations) or a vector of length n of standard deviations.

  Note that mean and sd

  must have the same length, either 1 or n.

• a: truncation limit 1 (if common for all observations) or a vector of length n of truncation limits 1.

• b: truncation limit 2 (if common for all observations) or a vector of length n of truncation limits 2.

• trunc: type of truncation (if common for all observations) or a vector of length n of types of truncation

  • trunc=0: normal distribution is truncated on the interval $(a, Infty)$. Value of $b$ is ignored.

  • trunc=1: degenerated normal distribution, all values are with probability 1 equal to $a$, $b$ is ignored.

  • trunc=2: normal distribution is truncated on the interval $(-Infty, a).$ Value of $b$ is ignored.

  • trunc=3: normal distribution is truncated on the interval $(a,\,b).$

  • trunc=4: there is no truncation, values of $a$

     and $b$ are ignored.
    

  If trunc is not given, it is assumed that it is equal to 4. Note that a, b and trunc

  must have the same length, either 1 or n with exception that b does not have to be supplied if trunc is 0, 1, 2 or 4.

• n: number of observations to be sampled.

Returns

A numeric vector with sampled values.

References

Geweke, J. (1991). Efficient simulation from the multivariate normal and Student-t distributions subject to linear constraints and the evaluation of constraint probabilities. Computer Sciences and Statistics, 23 , 571--578.

See Also

rnorm, rTMVN.

Author(s)

Arnošt Komárek arnost.komarek@mff.cuni.cz

Examples

set.seed(1977)

### Not truncated normal distribution
x1 <- rTNorm(1000, mean=10, sd=3)
c(mean(x1), sd(x1), range(x1))

### Truncation from left only
x2 <- rTNorm(1000, mean=10, sd=3, a=7, trunc=0)
c(mean(x2), sd(x2), range(x2))

### Degenerated normal distribution
x6 <- rTNorm(1000, mean=10, sd=3, a=13, trunc=1)
c(mean(x6), sd(x6), range(x6))

### Truncation from right only
x3 <- rTNorm(1000, mean=10, sd=3, a=13, trunc=2)
c(mean(x3), sd(x3), range(x3))

### Truncation from both sides
x4 <- rTNorm(1000, mean=10, sd=3, a=7, b=13, trunc=3)
c(mean(x4), sd(x4), range(x4))

x5 <- rTNorm(1000, mean=10, sd=3, a=5.5, b=14.5, trunc=3)
c(mean(x5), sd(x5), range(x5))

oldPar <- par(mfrow=c(2, 3))
hist(x1, main="N(10, 3^2)")
hist(x2, main="TN(10, 3^2, 7, Infty)")
hist(x6, main="TN(10, 3^2, 13, 13)")
hist(x3, main="TN(10, 3^2, -Infty, 13)")
hist(x4, main="TN(10, 3^2, 7, 13)")
hist(x5, main="TN(10, 3^2, 5.5, 14.5)")
par(oldPar)

### Different truncation limits
n <- 1000
a <- rnorm(n, -2, 1)
b <- a + rgamma(n, 1, 1)
trunc <- rep(c(0, 1, 2, 3, 4), each=n/5)
x7 <- rTNorm(n, mean=1, sd=2, a=a, b=b, trunc=trunc)
cbind(trunc, a, x7)[1:10,]
sum(x7[1:(n/5)] > a[1:(n/5)])      ## must be equal to n/5

cbind(trunc, a, x7)[201:210,]
sum(x7[(n/5+1):(2*n/5)] == a[(n/5+1):(2*n/5)])         ## must be equal to n/5

cbind(trunc, x7, a)[401:410,]
sum(x7[(2*n/5+1):(3*n/5)] < a[(2*n/5+1):(3*n/5)])      ## must be equal to n/5

cbind(trunc, a, x7, b)[601:610,]
sum(x7[(3*n/5+1):(4*n/5)] > a[(3*n/5+1):(4*n/5)])      ## must be equal to n/5
sum(x7[(3*n/5+1):(4*n/5)] < b[(3*n/5+1):(4*n/5)])      ## must be equal to n/5

cbind(trunc, x7)[801:810,]

### Different moments and truncation limits
n <- 1000
mu <- rnorm(n, 1, 0.2)
sigma <- 0.5 + rgamma(n, 1, 1)
a <- rnorm(n, -2, 1)
b <- a + rgamma(n, 1, 1)
trunc <- rep(c(0, 1, 2, 3, 4), each=n/5)
x8 <- rTNorm(n, mean=1, sd=2, a=a, b=b, trunc=trunc)

### Truncation from left only
### (extreme cases when we truncate to the area
###  where the original normal distribution has
###  almost zero probability)
x2b <- rTNorm(1000, mean=0, sd=1, a=7.9, trunc=0)
c(mean(x2b), sd(x2b), range(x2b))

x2c <- rTNorm(1000, mean=1, sd=2, a=16, trunc=0)
c(mean(x2c), sd(x2c), range(x2c))

### Truncation from right only (extreme cases)
x3b <- rTNorm(1000, mean=0, sd=1, a=-7.9, trunc=2)
c(mean(x3b), sd(x3b), range(x3b))

x3c <- rTNorm(1000, mean=1, sd=2, a=-13, trunc=2)
c(mean(x3c), sd(x3c), range(x3c))

### Truncation from both sides (extreme cases)
x4b <- rTNorm(1000, mean=0, sd=1, a=-9, b=-7.9, trunc=3)
c(mean(x4b), sd(x4b), range(x4b))

x4c <- rTNorm(1000, mean=0, sd=1, a=7.9, b=9, trunc=3)
c(mean(x4c), sd(x4c), range(x4c))