VGAM (version 1.1-8)

logistic: Logistic Distribution Family Function

Description

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

Usage

logistic1(llocation = "identitylink", scale.arg = 1, imethod = 1)
logistic(llocation = "identitylink", lscale = "loglink",
         ilocation = NULL, iscale = NULL, imethod = 1, zero = "scale")

Value

An object of class "vglmff"

(see vglmff-class). The object is used by modelling functions such as vglm,

rrvglm and vgam.

Arguments

llocation, lscale

Parameter link functions applied to the location parameter \(l\) and scale parameter \(s\). See Links for more choices, and CommonVGAMffArguments for more information.

scale.arg

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

ilocation, iscale

See CommonVGAMffArguments for information.

imethod, zero

See CommonVGAMffArguments for information.

Author

T. W. Yee

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 \(-\infty<y<\infty\). The mean of \(Y\) (which is the fitted value) is \(l\) and its variance is \(\pi^2 s^2 / 3\).

A logistic distribution with scale = 0.65 (see dlogis) resembles dt with df = 7; see logistic1 and studentt.

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

logistic can handle multiple responses.

References

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

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

Castillo, E., Hadi, A. S., Balakrishnan, N. and Sarabia, J. S. (2005). Extreme Value and Related Models with Applications in Engineering and Science, Hoboken, NJ, USA: 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, CommonVGAMffArguments, logitlink, cumulative, bilogistic, simulate.vlm.

Examples

# Location unknown, scale known
ldata <- data.frame(x2 = runif(nn <- 500))
ldata <- transform(ldata, y1 = rlogis(nn, loc = 1+5*x2, sc = exp(2)))
fit1 <- vglm(y1 ~ x2, logistic1(scale = exp(2)), ldata, trace = TRUE)
coef(fit1, matrix = TRUE)

# Both location and scale unknown
ldata <- transform(ldata, y2 = rlogis(nn, loc = 1 + 5*x2, exp(x2)))
fit2 <- vglm(cbind(y1, y2) ~ x2, logistic, data = ldata, trace = TRUE)
coef(fit2, matrix = TRUE)
vcov(fit2)
summary(fit2)