logistic: Logistic Distribution Family Function

In VGAM: Vector Generalized Linear and Additive Models

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")

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.

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.

Value

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

Note

Fisher scoring is used, and the Fisher information matrix is diagonal.

Author(s)

T. W. Yee

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, gensh, 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)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.