R: Positive Normal Distribution Family Function

posnormal {VGAM} | R Documentation

## Positive Normal Distribution Family Function

### Description

Fits a positive (univariate) normal distribution.

### Usage

    posnormal(lmean = "identitylink", lsd = "loge",
              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) = (1/(sqrt(2*pi)*sigma)) * exp( -0.5 * (y-mu)^2/ sigma^2) / [1-Phi(-mu/ sigma)]

where Phi is the cumulative distribution function of a standard normal (`pnorm`). Equivalently, this is

f(y) = (1/sigma) * dnorm((y-mu)/sigma) / [1-pnorm(-mu/ sigma)].

where dnorm() is the probability density function of a standard normal distribution (`dnorm`).

The mean of Y is

E(Y) = mu + sigma * dnorm((y-mu)/sigma) / [1-pnorm(-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`.

### 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.

### 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.

### Author(s)

Thomas W. Yee

### 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))

    ## Not run: with(pdata, hist(y, prob = TRUE, border = "blue",
      main = paste("posnorm(m =", Mean[1], ", sd =", round(SD[1], 2),")")))
    ## End(Not run)
    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))  # Add the fit to the histogram
    ## Not run: lines(mygrid, dposnorm(mygrid, Cfit[1], Cfit[2]), col = "orange")

[Package VGAM version 1.0-3 Index]