ExtDist (version 0.6-3)

Normal_trunc_ab: The truncated normal distribution.

Description

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.

Usage

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 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

See Also

ExtDist for other standard distributions.

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
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)$par.type=="boundary"]
est.par[attributes(est.par)$par.type=="shape"]

# log-likelihood function
lNormal_trunc_ab(X,param = est.par)