logff {VGAM} | R Documentation

## Logarithmic Distribution

### Description

Estimating the (single) parameter of the logarithmic distribution.

### Usage

    logff(link = "logit", earg=list(), init.c = NULL)

### Arguments

`link` | Parameter link function applied to the parameter c, which lies between 0 and 1. See `Links` for more choices.
`earg` | List. Extra argument for the link. See `earg` in `Links` for general information.
`init.c` | Optional initial value for the c parameter. If given, it often pays to start with a larger value, e.g., 0.95. The default is to choose an initial value internally.

### Details

The logarithmic distribution is based on the logarithmic series, and is scaled to a probability function. Its probability function is f(y) = a * c^y / y, for y=1,2,3,..., where 0 < c < 1, 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.

### 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 `loge` corresponds to this.

The logarithmic distribution is sometimes confused with the log-series distribution. The latter was used by Fisher et al. for species abundance data, and has two parameters.

### Author(s)

T. W. Yee

### References

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

Evans, M., Hastings, N. and Peacock, B. (2000) Statistical Distributions, New York: Wiley-Interscience, Third edition.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

### See Also

`rlog`, `log`, `loge`, `logoff`.

### Examples

    y = rlog(n=1000, prob=logit(0.2, inverse=TRUE))
    fit = vglm(y ~ 1, logff, trace=TRUE, crit="c")
    coef(fit, matrix=TRUE)
    Coef(fit)
    ## Not run:
    hist(y, prob=TRUE, breaks=seq(0.5, max(y)+0.5, by=1), border="blue")
    x = seq(1, max(y), by=1)
    lines(x, dlog(x, Coef(fit)[1]), col="red", type="h", lwd=2)
    ## End(Not run)

[Package VGAM version 0.7-9 Index]