logff: Logarithmic Distribution

In VGAM: Vector Generalized Linear and Additive Models

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 c, 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 c 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) = a c^y / y, for y=1,2,3,\ldots, where 0 < c < 1 (called shape), and a = -1 / \log(1-c). The mean is a c/(1-c) (returned as the fitted values) and variance is a c (1-ac) /(1-c)^2. When the sample mean is large, the value of c 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] 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)

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