VGAM (version 1.1-8)

logff: Logarithmic Distribution

Description

Estimating the (single) parameter of the logarithmic distribution.

Usage

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

Value

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

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.

Author

T. W. Yee

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.

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)
if (FALSE) 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)) 

# 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)