bilogistic {VGAM} | R Documentation

## Bivariate Logistic Distribution Family Function

### Description

Estimates the four parameters of the bivariate logistic distribution by maximum likelihood estimation.

### Usage

    bilogistic(llocation = "identitylink", lscale = "loge",
               iloc1 = NULL, iscale1 = NULL, iloc2 = NULL, iscale2 = NULL,
               imethod = 1, zero = NULL)

### Arguments

`llocation` | Link function applied to both location parameters l1 and l2. See `Links` for more choices.
`lscale` | Parameter link function applied to both (positive) scale parameters s1 and s2. See `Links` for more choices.
`iloc1, iloc2` | Initial values for the location parameters. By default, initial values are chosen internally using `imethod`. Assigning values here will override the argument `imethod`.
`iscale1, iscale2` | Initial values for the scale parameters. By default, initial values are chosen internally using `imethod`. Assigning values here will override the argument `imethod`.
`imethod` | 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, one can choose values from the set {1,2,3,4}. See `CommonVGAMffArguments` for more information.

### Details

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

f(y1,y2;l1,s1,l2,s2) = 2 * exp[-(y1-l1)/s1 - (y1-l1)/s1] / [s1 * s2 * ( 1 + exp[-(y1-l1)/s1] + exp[-(y2-l2)/s2] )^3]

where s1>0 and s2>0 are the scale parameters, and l1 and l2 are the location parameters. Each of the two responses are unbounded, i.e., -Inf<y_j<Inf. The mean of Y1 is l1 etc. The fitted values are returned in a 2-column matrix. The cumulative distribution function is

F(y1,y2;l1,s1,l2,s2) = 1 / (1 + exp[-(y1-l1)/s1] + exp[-(y2-l2)/s2])

The marginal distribution of Y1 is

P(Y1 <= y1) = F(y1;l1,s1) = 1 / (1 + exp[-(y1-l1)/s1]).

By default, eta1=l1, eta2=log(s1), eta3=l2, eta4=log(s2) are the linear/additive predictors.

### Value

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

### Note

This family function uses the BFGS quasi-Newton update formula for the working weight matrices. Consequently the estimated variance-covariance matrix may be inaccurate or simply wrong! The standard errors must be therefore treated with caution; these are computed in functions such as `vcov()` and `summary()`.

### Author(s)

T. W. Yee

### References

Gumbel, E. J. (1961) Bivariate logistic distributions. Journal of the American Statistical Association, 56, 335–349.

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

### See Also

`logistic`, `rbilogis`.

### Examples

    ymat <- rbilogis(n <- 1000, loc1 = 5, loc2 = 7, scale2 = exp(1))
    ## Not run: plot(ymat)
    fit <- vglm(ymat ~ 1, fam = bilogistic, trace = TRUE)
    coef(fit, matrix = TRUE)
    Coef(fit)
    head(fitted(fit))
    vcov(fit)
    head(weights(fit, type = "work"))
    summary(fit)

[Package VGAM version 1.0-4 Index]