# 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`. ```r 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 ```r 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)) ```

TNorm function