logff {VGAM}R Documentation

Logarithmic Distribution

Description

Estimating the (single) parameter of the logarithmic distribution.

Usage

logff(lshape = "logitlink", gshape = -expm1(-7 * ppoints(4)),
      zero = NULL)

Arguments

lshape

Parameter link function for the parameter cc, which lies between 0 and 1. See Links for more choices and information. Soon logfflink() will hopefully be available for event-rate data.

gshape, zero

Details at CommonVGAMffArguments. Practical experience shows that having the initial value for cc being close to the solution is quite important.

Details

The logarithmic distribution is a generalized power series distribution that is based specifically on the logarithmic series (scaled to a probability function). Its probability function is f(y)=acy/yf(y) = a c^y / y, for y=1,2,3,y=1,2,3,\ldots, where 0<c<10 < c < 1 (called shape), and a=1/log(1c)a = -1 / \log(1-c). The mean is ac/(1c)a c/(1-c) (returned as the fitted values) and variance is ac(1ac)/(1c)2a c (1-ac) /(1-c)^2. When the sample mean is large, the value of cc tends to be very close to 1, hence it could be argued that logitlink is not the best choice.

Value

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

Note

The function log computes the natural logarithm. In the VGAM library, a link function with option loglink corresponds to this.

Multiple responses are permitted.

The “logarithmic distribution” has various meanings in the literature. Sometimes it is also called the log-series distribution. Some others call some continuous distribution on [a,b][a, b] by the name “logarithmic distribution”.

Author(s)

T. W. Yee

References

Johnson N. L., Kemp, A. W. and Kotz S. (2005). Univariate Discrete Distributions, 3rd edition, ch.7. Hoboken, New Jersey: Wiley.

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

See Also

Log, gaitdlog, oalog, oilog, otlog, log, loglink, logofflink, explogff, simulate.vlm.

Examples

nn <- 1000
ldata <- data.frame(y = rlog(nn, shape = logitlink(0.2, inv = TRUE)))
fit <- vglm(y ~ 1, logff, data = ldata, trace = TRUE, crit = "c")
coef(fit, matrix = TRUE)
Coef(fit)
## Not run: with(ldata, spikeplot(y, col = "blue", capped = TRUE))
x <- seq(1, with(ldata, max(y)), by = 1)
with(ldata, lines(x + 0.1, dlog(x, Coef(fit)[1]), col = "orange",
        type = "h", lwd = 2)) 
## End(Not run)

# Example: Corbet (1943) butterfly Malaya data
corbet <- data.frame(nindiv = 1:24,
                 ofreq = c(118, 74, 44, 24, 29, 22, 20, 19, 20, 15, 12,
                           14, 6, 12, 6, 9, 9, 6, 10, 10, 11, 5, 3, 3))
fit <- vglm(nindiv ~ 1, logff, data = corbet, weights = ofreq)
coef(fit, matrix = TRUE)
shapehat <- Coef(fit)["shape"]
pdf2 <- dlog(x = with(corbet, nindiv), shape = shapehat)
print(with(corbet, cbind(nindiv, ofreq, fitted = pdf2 * sum(ofreq))),
      digits = 1)

[Package VGAM version 1.1-11 Index]