R: Logarithmic Distribution

logff {VGAM} | R Documentation

## Logarithmic Distribution

### Description

Estimating the (single) parameter of the logarithmic distribution.

### Usage

    logff(lshape = "logit", gshape = ppoints(8), 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`.

### 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.

### 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.

Multiple responses are permitted.

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.

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011) Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

### See Also

`rlog`, `oalog`, `oilog`, `otlog`, `log`, `loge`, `logoff`, `explogff`, `simulate.vlm`.

### Examples

    ldata <- data.frame(y = rlog(n = 1000, shape = logit(0.2, inverse = 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)

[Package VGAM version 1.0-3 Index]