VGAM (version 0.8-1)

logistic: Logistic Distribution Family Function

Description

Estimates the location and scale parameters of the logistic distribution by maximum likelihood estimation.

Usage

logistic1(llocation="identity", elocation=list(),
          scale.arg=1, method.init=1)
logistic2(llocation="identity", lscale="loge",
          elocation=list(), escale=list(),
          ilocation=NULL, iscale=NULL, method.init=1, zero=NULL)

Arguments

llocation

Link function applied to the location parameter $l$. See Links for more choices.

elocation, escale

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

scale.arg

Known positive scale parameter (called $s$ below).

lscale

Parameter link function applied to the scale parameter $s$. See Links for more choices.

ilocation

Initial value for the location $l$ parameter. By default, an initial value is chosen internally using method.init. Assigning a value will override the argument method.init.

iscale

Initial value for the scale $s$ parameter. By default, an initial value is chosen internally using method.init. Assigning a value will override the argument method.init.

method.init

An integer with value 1 or 2 which specifies the initialization method. If failure to converge occurs try the other value.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The default is none of them. If used, choose one value from the set {1,2}.

Value

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

Details

The two-parameter logistic distribution has a density that can be written as $$f(y;l,s) = \frac{\exp[-(y-l)/s]}{ s\left( 1 + \exp[-(y-l)/s] \right)^2}$$ where $s>0$ is the scale parameter, and $l$ is the location parameter. The response $-\infty

logistic1 estimates the location parameter only while logistic2 estimates both parameters. By default, $\eta_1=l$ and $\eta_2=\log(s)$ for logistic2.

References

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, 2nd edition, Volume 1, New York: Wiley. Chapter 15.

Evans, M., Hastings, N. and Peacock, B. (2000) Statistical Distributions, New York: Wiley-Interscience, Third edition.

Castillo, E., Hadi, A. S., Balakrishnan, N. Sarabia, J. S. (2005) Extreme Value and Related Models with Applications in Engineering and Science, Hoboken, N.J.: Wiley-Interscience, p.130.

deCani, J. S. and Stine, R. A. (1986) A note on Deriving the Information Matrix for a Logistic Distribution, The American Statistician, 40, 220--222.

See Also

rlogis, logit, cumulative, bilogistic4.

Examples

# location unknown, scale known
lodat1 = data.frame(x = runif(nn <- 500))
lodat1 = transform(lodat1, y = rlogis(nn, loc=1+5*x, scale=4))
fit = vglm(y ~ x, logistic1(scale=4), lodat1, trace=TRUE, crit="c")
coef(fit, matrix=TRUE)

# Both location and scale unknown
lodat2 = data.frame(x = runif(nn <- 2000))
lodat2 = transform(lodat2, y = rlogis(nn, loc=1+5*x, scale=exp(0+1*x)))
fit = vglm(y ~ x, logistic2, lodat2)
coef(fit, matrix=TRUE)
vcov(fit)
summary(fit)