VGAM (version 1.1-11)

lognormal: Lognormal Distribution

Description

Maximum likelihood estimation of the (univariate) lognormal distribution.

Usage

lognormal(lmeanlog = "identitylink", lsdlog = "loglink", zero = "sdlog")

Value

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

Arguments

lmeanlog, lsdlog

Parameter link functions applied to the mean and (positive) $\sigma$ (standard deviation) parameter. Both of these are on the log scale. See Links for more choices.

zero

Specifies which linear/additive predictor is modelled as intercept-only. For lognormal(), the values can be from the set {1,2} which correspond to mu, sigma, respectively. See CommonVGAMffArguments for more information.

Author

T. W. Yee

Details

A random variable $Y$ has a 2-parameter lognormal distribution if $\log (Y)$ is distributed $N(\mu ,{\sigma }^2)$. The expected value of $Y$, which is $$E(Y)=\exp (\mu +0.5{\sigma }^2)$$ and not $\mu$, make up the fitted values. The variance of $Y$ is $$Var(Y)=[\exp ({\sigma }^2)-1]\exp (2\mu +{\sigma }^2).$$

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

See Also

Lognormal, uninormal, CommonVGAMffArguments, simulate.vlm.

Examples

ldata2 <- data.frame(x2 = runif(nn <- 1000))
ldata2 <- transform(ldata2, y1 = rlnorm(nn, 1 + 2 * x2, sd = exp(-1)),
                            y2 = rlnorm(nn, 1, sd = exp(-1 + x2)))
fit1 <- vglm(y1 ~ x2, lognormal(zero = 2), data = ldata2, trace = TRUE)
fit2 <- vglm(y2 ~ x2, lognormal(zero = 1), data = ldata2, trace = TRUE)
coef(fit1, matrix = TRUE)
coef(fit2, matrix = TRUE)