The truncated normal distribution.

Density, distribution, quantile, random number generation and parameter estimation functions for the truncated normal distribution with parameters mean, sd and a and b which represent the lower and upper truncation points respectively. Parameter estimation can be based on a weighted or unweighted i.i.d. sample and is performed numerically.

dNormal_trunc_ab(
  x,
  mu = 0,
  sigma = 1,
  a = 0,
  b = 1,
  params = list(mu, sigma, a, b),
  ...
)

pNormal_trunc_ab(
  q,
  mu = 0,
  sigma = 1,
  a = 0,
  b = 1,
  params = list(mu = 2, sigma = 5, a = 0, b = 1),
  ...
)

qNormal_trunc_ab(
  p,
  mu = 0,
  sigma = 1,
  a = 0,
  b = 1,
  params = list(mu = 2, sigma = 5, a = 0, b = 1),
  ...
)

rNormal_trunc_ab(
  n,
  mu = 0,
  sigma = 1,
  a = 0,
  b = 1,
  params = list(mu, sigma, a, b),
  ...
)

eNormal_trunc_ab(X, w, method = "numerical.MLE", ...)

lNormal_trunc_ab(
  X,
  w,
  mu = 0,
  sigma = 1,
  a = 0,
  b = 1,
  params = list(mu, sigma, a, b),
  logL = TRUE,
  ...
)

Arguments

x, q: A vector of quantiles.
mu, sigma: Shape parameters.
a, b: Boundary parameters.
params: A list that includes all named parameters.
...: Additional parameters.
p: A vector of probabilities.
n: Number of observations.
X: Sample observations.
w: An optional vector of sample weights.
method: Parameter estimation method.
logL: logical;if TRUE, lNormal_trunc_ab gives the log-likelihood, otherwise the likelihood is given.

Value

dNormal_trunc_ab gives the density, pNormal_trunc_ab the distribution function, qNormal_trunc_ab the quantile function, rNormal_trunc_ab generates random variables, and eNormal_trunc_ab estimates the parameters. lNormal_trunc_ab provides the log-likelihood function.

Details

If the mean, sd, a or b are not specified they assume the default values of 0, 1, 0, 1 respectively.

The dNormal_trunc_ab(), pNormal_trunc_ab(), qNormal_trunc_ab(),and rNormal_trunc_ab() functions serve as wrappers of the dtrunc, ptrunc, qtrunc, and rtrunc functions in the truncdist package. They allow for the parameters to be declared not only as individual numerical values, but also as a list so parameter estimation can be carried out.

The probability density function of the doubly truncated normal distribution is given by $$f(x) = \sigma^{-1} Z(x-\mu/\sigma)[\Phi(b-\mu/\sigma) - \Phi(a-\mu/\sigma)]^{-1}$$ where $\infty <a \le x \le b < \infty$. The degrees of truncation are $\Phi((a-\mu)/\sigma)$ from below and $1-\Phi((a-\mu)/\sigma)$ from above. If a is replaced by $-\infty$, or b by $\infty$, the distribution is singly truncated, (Johnson et.al, p.156). The upper and lower limits of truncation $a$ and $b$ are normally known parameters whereas $\mu$ and $\sigma$ may be unknown. Crain (1979) discusses parameter estimation for the truncated normal distribution and the method of numerical maximum likelihood estimation is used for parameter estimation in eNormal_trunc_ab.

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, volume 1, chapter 13, Wiley, New York.

Crain, B.R (1979). Estimating the parameters of a truncated normal distribution, Applied Mathematics and Computation, vol 4, pp. 149-156

Author

Haizhen Wu and A. Jonathan R. Godfrey.
Updates and bug fixes by Sarah Pirikahu.

Examples

# Parameter estimation for a distribution with known shape parameters
X <- rNormal_trunc_ab(n= 500, mu= 2, sigma = 5, a = 1, b = 2)
est.par <- eNormal_trunc_ab(X); est.par
#> 
#> Parameters for the Normal_trunc_ab distribution. 
#> (found using the  numerical.MLE method.)
#> 
#>  Parameter     Type Estimate
#>         mu    shape 1.539292
#>      sigma    shape 1.172889
#>          a boundary 1.001297
#>          b boundary 1.999830
#> 
#> 
plot(est.par)


#  Fitted density curve and histogram
den.x <- seq(min(X),max(X),length=100)
den.y <- dNormal_trunc_ab(den.x,params = est.par)
hist(X, breaks=10, probability=TRUE, ylim = c(0,1.2*max(den.y)))
lines(den.x, den.y, col="blue")
lines(density(X), lty = 2)


# Extracting boundary and shape parameters
est.par[attributes(est.par)$par.type=="boundary"]
#> $a
#> [1] 1.001297
#> 
#> $b
#> [1] 1.99983
#> 
est.par[attributes(est.par)$par.type=="shape"]
#> $mu
#> [1] 1.539292
#> 
#> $sigma
#> [1] 1.172889
#> 

# log-likelihood function
lNormal_trunc_ab(X,param = est.par)
#> [1] 0.9300112