# Logarithmic Distribution Estimating the (single) parameter of the logarithmic distribution. ```r 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 $y=1,2,3,...$, 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. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`. ## 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. ## Author(s) T. W. Yee ## 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''. ## See Also `Log`, `gaitdlog`, `oalog`, `oilog`, `otlog`, `log`, `loglink`, `logofflink`, `explogff`, `simulate.vlm`. ## Examples ```r 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) ```

logff function