Logarithmic Distribution
Estimating the (single) parameter of the logarithmic distribution.
logff(lshape = "logitlink", gshape = -expm1(-7 * ppoints(4)), zero = NULL)
lshape |
Parameter link function for the parameter c,
which lies between 0 and 1.
See |
gshape, zero |
Details at |
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,...,
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-a*c)/(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.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
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”.
T. W. Yee
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.
Log,
gaitlog,
oalog,
oilog,
otlog,
log,
loglink,
logofflink,
explogff,
simulate.vlm.
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,
hist(y, prob = TRUE, breaks = seq(0.5, max(y) + 0.5, by = 1),
border = "blue"))
x <- seq(1, with(ldata, max(y)), by = 1)
with(ldata, lines(x, 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.