VGAM (version 0.9-1)

logistic: Logistic Distribution Family Function

Description

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

Usage

logistic1(llocation = "identity", scale.arg = 1, imethod = 1)
logistic2(llocation = "identity", lscale = "loge",
          ilocation = NULL, iscale = NULL, imethod = 1, zero = -2)

Arguments

llocation, lscale

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

scale.arg

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

ilocation, iscale

See CommonVGAMffArguments for more information.

imethod, zero

See CommonVGAMffArguments for more information.

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

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 logistic2 estimates both parameters. By default, $\eta_1 = l$ and $\eta_2 = \log(s)$ for logistic2.

logistic2 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.

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, 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, logit, cumulative, bilogistic4.

Examples

# Location unknown, scale known
ldata <- data.frame(x2 = runif(nn <- 500))
ldata <- transform(ldata, y1 = rlogis(nn, loc = 1+5*x2, scale = exp(2)))
fit1 <- vglm(y1 ~ x2, logistic1(scale = 4), ldata, trace = TRUE, crit = "c")
coef(fit1, matrix = TRUE)

# Both location and scale unknown
ldata <- transform(ldata, y2 = rlogis(nn, loc = 1+5*x2, scale = exp(0+1*x2)))
fit2 <- vglm(cbind(y1, y2) ~ x2, logistic2, ldata, trace = TRUE)
coef(fit2, matrix = TRUE)
vcov(fit2)
summary(fit2)