Logistic Distribution Family Function
Estimates the location and scale parameters of the logistic distribution by maximum likelihood estimation.
logistic1(llocation = "identitylink", scale.arg = 1, imethod = 1)
logistic(llocation = "identitylink", lscale = "loglink",
ilocation = NULL, iscale = NULL, imethod = 1, zero = "scale")llocation, lscale |
Parameter link functions applied to the location parameter l
and scale parameter s.
See |
scale.arg |
Known positive scale parameter (called s below). |
ilocation, iscale |
See |
imethod, zero |
See |
The two-parameter logistic distribution has a density that can be written as
f(y;l,s) = exp[-(y-l)/s] / [s * ( 1 + exp[-(y-l)/s] )^2]
where s > 0 is the scale parameter, and l is the location parameter. The response -Inf<y<Inf. The mean of Y (which is the fitted value) is l and its variance is pi^2 s^2 / 3.
logistic1 estimates the location parameter only while
logistic estimates both parameters.
By default,
eta1 = l and eta2 = log(s) for
logistic.
logistic can handle multiple responses.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
rrvglm and vgam.
Fisher scoring is used, and the Fisher information matrix is diagonal.
T. W. Yee
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. 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.
# 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 = exp(2)), data = ldata, trace = TRUE) 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, logistic, data = ldata, trace = TRUE) coef(fit2, matrix = TRUE) vcov(fit2) summary(fit2)
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