VGAM (version 1.1-14)

posnormal: Positive Normal Distribution Family Function

Description

Fits a positive (univariate) normal distribution.

Usage

posnormal(lmean = "identitylink", lsd = "loglink",
          eq.mean = FALSE, eq.sd = FALSE,
          gmean = exp((-5:5)/2), gsd = exp((-1:5)/2),
          imean = NULL, isd = NULL, probs.y = 0.10, imethod = 1,
          nsimEIM = NULL, zero = "sd")

Value

An object of class "vglmff"

(see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Details

The positive normal distribution is the ordinary normal distribution but with the probability of zero or less being zero. The rest of the probability density function is scaled up. Hence the probability density function can be written $$f(y) = \frac{1}{\sqrt{2\pi} \sigma} \exp\left( -\frac12 (y-\mu)^2 / \sigma^2 \right) / \left[ 1-\Phi(-\mu/ \sigma) \right]$$ where \(\Phi()\) is the cumulative distribution function of a standard normal (pnorm). Equivalently, this is $$f(y) = \frac{1}{\sigma} \frac{\phi((y-\mu) / \sigma)}{ 1-\Phi(-\mu/ \sigma)}$$ where \(\phi()\) is the probability density function of a standard normal distribution (dnorm).

The mean of \(Y\) is $$E(Y) = \mu + \sigma \frac{\phi(-\mu/ \sigma)}{ 1-\Phi(-\mu/ \sigma)}. $$ This family function handles multiple responses.

See Also

uninormal, tobit.

Examples

pdata <- data.frame(Mean = 1.0, SD = exp(1.0))
pdata <- transform(pdata, y = rposnorm(n <- 1000, m = Mean, sd = SD))

if (FALSE) with(pdata, hist(y, prob = TRUE, border = "blue",
  main = paste("posnorm(m =", Mean[1], ", sd =", round(SD[1], 2),")")))
fit <- vglm(y ~ 1, posnormal, data = pdata, trace = TRUE)
coef(fit, matrix = TRUE)
(Cfit <- Coef(fit))
mygrid <- with(pdata, seq(min(y), max(y), len = 200))
if (FALSE) lines(mygrid, dposnorm(mygrid, Cfit[1], Cfit[2]), col = "red")