erlang {VGAM} | R Documentation

## Erlang Distribution

### Description

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

### Usage

    erlang(shape.arg, link = "loge", earg=list(), method.init = 1)

### Arguments

`shape.arg` | The shape parameter. The user must specify a positive integer.
`link` | Link function applied to the (positive) scale parameter. See `Links` for more choices.
`earg` | List. Extra argument for the link. See `earg` in `Links` for general information.
`method.init` | An integer with value `1` or `2` which specifies the initialization method. If failure to converge occurs try the other value.

### 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

The `rate` parameter found in `gamma2.ab` is `1/scale` here—see also `rgamma`.

### Author(s)

T. W. Yee

### References

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

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

### See Also

`gamma2.ab`, `exponential`.

### Examples

    n = 200; rate = 8
    y = rep(0, n)
    for(i in 1:3)
        y = y + rexp(n, rate=rate)
    fit = vglm(y ~ 1, erlang(shape=3), trace=TRUE)
    coef(fit, matrix=TRUE)
    Coef(fit) # Answer = 1/rate
    1/rate
    summary(fit)

[Package VGAM version 0.7-9 Index]