erlang {VGAM} | R Documentation

## Erlang Distribution

### Description

Estimates the scale parameter of the Erlang distribution by maximum likelihood estimation.

### Usage

    erlang(shape.arg, lscale = "loge", imethod = 1, zero = NULL)

### Arguments

`shape.arg` | The shape parameters. The user must specify a positive integer, or integers for multiple responses. They are recycled `by.row = TRUE` according to `matrix`.
`lscale` | Link function applied to the (positive) scale parameter. See `Links` for more choices.
`imethod, zero` | See `CommonVGAMffArguments` for more details.

### Details

The Erlang distribution is a special case of the gamma distribution with shape that is a positive integer. If `shape.arg = 1` then it simplifies to the exponential distribution. As illustrated in the example below, the Erlang distribution is the distribution of the sum of `shape.arg` independent and identically distributed exponential random variates.

The probability density function of the Erlang distribution is given by

f(y) = exp(-y/scale) y^(shape-1) scale^(-shape) / gamma(shape)

for known positive integer shape, unknown scale > 0 and y > 0. Here, gamma(shape) is the gamma function, as in `gamma`. The mean of Y is mu=shape*scale and its variance is shape*scale^2. The linear/additive predictor, by default, is eta=log(scale).

### Value

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

### Note

Multiple responses are permitted. The `rate` parameter found in `gammaR` is `1/scale` here—see also `rgamma`.

### Author(s)

T. W. Yee

### References

Most standard texts on statistical distributions describe this distribution, e.g.,

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

### See Also

`gammaR`, `exponential`, `simulate.vlm`.

### Examples

    rate <- exp(2); myshape <- 3
    edata <- data.frame(y = rep(0, nn <- 1000))
    for (ii in 1:myshape)
      edata <- transform(edata, y = y + rexp(nn, rate = rate))
    fit <- vglm(y ~ 1, erlang(shape = myshape), data = edata, trace = TRUE)
    coef(fit, matrix = TRUE)
    Coef(fit)  # Answer = 1/rate
    1/rate
    summary(fit)

[Package VGAM version 1.0-3 Index]