Positive Normal Distribution Family Function

Fits a positive (univariate) normal distribution.

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")

Arguments

lmean, lsd

Link functions for the mean and standard deviation parameters of the usual univariate normal distribution. They are $\mu$ and $\sigma$ respectively. See Links for more choices.

gmean, gsd, imethod

See CommonVGAMffArguments for more information. gmean and gsd currently operate on a multiplicative scale, on the sample mean and the sample standard deviation, respectively.

imean, isd

Optional initial values for $\mu$ and $\sigma$. A NULL means a value is computed internally. See CommonVGAMffArguments for more information.

eq.mean, eq.sd

See CommonVGAMffArguments for more information. The fact that these arguments are supported results in default constraint matrices being a permutation of the identity matrix (effectively trivial constraints).

zero, nsimEIM, probs.y

See CommonVGAMffArguments for information.

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.

Value

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

Author

Thomas W. Yee

Note

The response variable for this family function is the same as uninormal except positive values are required. Reasonably good initial values are needed.

The distribution of the reciprocal of a positive normal random variable is known as an alpha distribution.

Warning

It is recommended that trace = TRUE be used to monitor convergence; sometimes the estimated mean is -Inf and the estimated mean standard deviation is Inf, especially when the sample size is small. Under- or over-flow may occur if the data is ill-conditioned.

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),")"))) # \dontrun{}
fit <- vglm(y ~ 1, posnormal, data = pdata, trace = TRUE)
#> Iteration 1: loglikelihood = -1861.6716
#> Iteration 2: loglikelihood = -1861.2365
#> Iteration 3: loglikelihood = -1861.2363
#> Iteration 4: loglikelihood = -1861.2363
coef(fit, matrix = TRUE)
#>                 mean loglink(sd)
#> (Intercept) 1.217719   0.9496246
(Cfit <- Coef(fit))
#>     mean       sd 
#> 1.217719 2.584739 
mygrid <- with(pdata, seq(min(y), max(y), len = 200))
if (FALSE) lines(mygrid, dposnorm(mygrid, Cfit[1], Cfit[2]), col = "red") # \dontrun{}