logistic {VGAM} | R Documentation

## Logistic Distribution Family Function

### Description

Estimates the location and scale parameters of the logistic distribution by maximum likelihood estimation.

### Usage

    logistic1(llocation="identity", elocation=list(),
              scale.arg=1, method.init=1)
    logistic2(llocation="identity", lscale="loge",
              elocation=list(), escale=list(),
              ilocation=NULL, iscale=NULL, method.init=1, zero=NULL)

### Arguments

`llocation` | Link function applied to the location parameter l. See `Links` for more choices.
`elocation, escale` | List. Extra argument for each of the links. See `earg` in `Links` for general information.
`scale.arg` | Known positive scale parameter (called s below).
`lscale` | Parameter link function applied to the scale parameter s. See `Links` for more choices.
`ilocation` | Initial value for the location l parameter. By default, an initial value is chosen internally using `method.init`. Assigning a value will override the argument `method.init`.
`iscale` | Initial value for the scale s parameter. By default, an initial value is chosen internally using `method.init`. Assigning a value will override the argument `method.init`.
`method.init` | An integer with value `1` or `2` which specifies the initialization method. If failure to converge occurs try the other value.
`zero` | An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The default is none of them. If used, choose one value from the set {1,2}.

### Details

The two-parameter logistic distribution has a density that can be written as

f(y;l,s) = exp[-(y-l)/s] / [s * ( 1 + exp[-(y-l)/s] )^2]

where s>0 is the scale parameter, and l is the location parameter. The response -Inf<y<Inf. The mean of Y (which is the fitted value) is l and its variance is pi^2 s^2 / 3.

`logistic1` estimates the location parameter only while `logistic2` estimates both parameters. By default, eta1=l and eta2=log(s) for `logistic2`.

### Value

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

### Note

Fisher scoring is used, and the Fisher information matrix is diagonal.

### Author(s)

T. W. Yee

### References

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, 2nd edition, Volume 1, New York: Wiley. Chapter 15.

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

Castillo, E., Hadi, A. S., Balakrishnan, N. Sarabia, J. S. (2005) Extreme Value and Related Models with Applications in Engineering and Science, Hoboken, N.J.: Wiley-Interscience, p.130.

deCani, J. S. and Stine, R. A. (1986) A note on Deriving the Information Matrix for a Logistic Distribution, The American Statistician, 40, 220–222.

### See Also

`rlogis`, `logit`, `cumulative`, `bilogistic4`.

### Examples

    # location unknown, scale known
    n = 500
    x = runif(n)
    y = rlogis(n, loc=1+5*x, scale=4)
    fit = vglm(y ~ x, logistic1(scale=4), trace=TRUE, crit="c")
    coef(fit, matrix=TRUE)

    # Both location and scale unknown
    n = 2000
    x = runif(n)
    y = rlogis(n, loc=1+5*x, scale=exp(0+1*x))
    fit = vglm(y ~ x, logistic2)
    coef(fit, matrix=TRUE)
    vcov(fit)
    summary(fit)

[Package VGAM version 0.7-9 Index]