VGAM (version 0.8-1)

lognormal: Lognormal Distribution

Description

Maximum likelihood estimation of the (univariate) lognormal distribution.

Usage

lognormal(lmeanlog = "identity", lsdlog = "loge",
          emeanlog=list(), esdlog=list(), zero = NULL)
lognormal3(lmeanlog = "identity", lsdlog = "loge",
           emeanlog=list(), esdlog=list(),
           powers.try = (-3):3, delta = NULL, zero = NULL)

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.

emeanlog, esdlog

List. Extra argument for each of the links. See earg in Links for general information.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. For lognormal(), the values must be from the set {1,2} which correspond to mu, sigma, respectively. For

powers.try

Numerical vector. The initial $lambda$ is chosen as the best value from min(y) - 10^powers.try where y is the response.

delta

Numerical vector. An alternative method for obtaining an initial $lambda$. Here, delta = min(y)-lambda. If given, this supersedes the powers.try argument. The value must be positive.

Value

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

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.

A random variable $Y$ has a 3-parameter lognormal distribution if $\log(Y-\lambda)$ is distributed $N(\mu, \sigma^2)$. Here, $\lambda < Y$. The expected value of $Y$, which is $$E(Y) = \lambda + \exp(\mu + 0.5 \sigma^2)$$ and not $\mu$, make up the fitted values.

lognormal() and lognormal3() fit the 2- and 3-parameter lognormal distribution respectively. Clearly, if the location parameter $\lambda=0$ then both distributions coincide.

References

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

See Also

rlnorm, normal1.

Examples

lndat = data.frame(y = rlnorm(nn <- 1000, meanlog=1.5, sdlog=exp(-0.8)))
fit = vglm(y ~ 1, lognormal, lndat, trace=TRUE)
coef(fit, mat=TRUE)
Coef(fit)

lndat2 = data.frame(x = runif(nn <- 1000))
lndat2 = transform(lndat2, y = rlnorm(nn, mean=0.5, sd=exp(x)))
fit = vglm(y ~ x, lognormal(zero=1), lndat2, trace=TRUE, crit="c")
coef(fit, mat=TRUE)
Coef(fit)

lambda = 4
lndat3 = data.frame(y = lambda + rlnorm(n=1000, mean=1.5, sd=exp(-0.8)))
fit = vglm(y ~ 1, lognormal3, lndat3, trace=TRUE, crit="c")
coef(fit, mat=TRUE)
summary(fit)