# bilogistic: Bivariate Logistic Distribution Family Function In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2 3``` ```bilogistic(llocation = "identitylink", lscale = "loglink", iloc1 = NULL, iscale1 = NULL, iloc2 = NULL, iscale2 = NULL, imethod = 1, nsimEIM = 250, 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. `nsimEIM, zero` See `CommonVGAMffArguments` for details.

## 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`.

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.

`logistic`, `rbilogis`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```## Not run: ymat <- rbilogis(n <- 50, loc1 = 5, loc2 = 7, scale2 = exp(1)) plot(ymat) bfit <- vglm(ymat ~ 1, family = bilogistic, trace = TRUE) coef(bfit, matrix = TRUE) Coef(bfit) head(fitted(bfit)) vcov(bfit) head(weights(bfit, type = "work")) summary(bfit) ## End(Not run) ```

### Example output

```sh: 1: cannot create /dev/null: Permission denied
VGLM    linear loop  1 :  loglikelihood = -504.74745
VGLM    linear loop  2 :  loglikelihood = -394.67814
VGLM    linear loop  3 :  loglikelihood = -363.09389
VGLM    linear loop  4 :  loglikelihood = -339.15823
VGLM    linear loop  5 :  loglikelihood = -315.99967
VGLM    linear loop  6 :  loglikelihood = -294.56885
VGLM    linear loop  7 :  loglikelihood = -277.30832
VGLM    linear loop  8 :  loglikelihood = -267.84732
VGLM    linear loop  9 :  loglikelihood = -265.32174
VGLM    linear loop  10 :  loglikelihood = -265.09976
VGLM    linear loop  11 :  loglikelihood = -265.09285
VGLM    linear loop  12 :  loglikelihood = -265.09268
VGLM    linear loop  13 :  loglikelihood = -265.09268
VGLM    linear loop  14 :  loglikelihood = -265.09268
(Intercept)  4.973317     -0.05816074  6.870092       0.8759847
location1    scale1 location2    scale2
4.9733174 0.9434983 6.8700920 2.4012386
y1          y2
1 4.973317 -0.05816074
2 4.973317 -0.05816074
3 4.973317 -0.05816074
4 4.973317 -0.05816074
5 4.973317 -0.05816074
6 4.973317 -0.05816074
(Intercept):1 (Intercept):2 (Intercept):3 (Intercept):4
(Intercept):1   0.049107537   0.002542029   0.056383024  -0.002551049
(Intercept):2   0.002542029   0.010768418  -0.006439701   0.002692852
(Intercept):3   0.056383024  -0.006439701   0.318090449   0.006463308
(Intercept):4  -0.002551049   0.002692852   0.006463308   0.010778829
dlocat1.deta dscale1.deta dlocat2.deta dscale2.deta dlocat1.deta dscale1.deta
1    0.5613184     2.147254   0.08663092     2.145802   -0.2647238    0.1038326
2    0.5613184     2.147254   0.08663092     2.145802   -0.2647238    0.1038326
3    0.5613184     2.147254   0.08663092     2.145802   -0.2647238    0.1038326
4    0.5613184     2.147254   0.08663092     2.145802   -0.2647238    0.1038326
5    0.5613184     2.147254   0.08663092     2.145802   -0.2647238    0.1038326
6    0.5613184     2.147254   0.08663092     2.145802   -0.2647238    0.1038326
dlocat2.deta dlocat1.deta dscale1.deta dlocat1.deta
1    -0.103978    -0.110242   -0.6613577    0.2650882
2    -0.103978    -0.110242   -0.6613577    0.2650882
3    -0.103978    -0.110242   -0.6613577    0.2650882
4    -0.103978    -0.110242   -0.6613577    0.2650882
5    -0.103978    -0.110242   -0.6613577    0.2650882
6    -0.103978    -0.110242   -0.6613577    0.2650882

Call:
vglm(formula = ymat ~ 1, family = bilogistic, trace = TRUE)

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1  4.97332    0.22160  22.443   <2e-16 ***
(Intercept):2 -0.05816    0.10377  -0.560    0.575
(Intercept):3  6.87009    0.56400  12.181   <2e-16 ***
(Intercept):4  0.87598    0.10382   8.437   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Names of linear predictors: location1, loglink(scale1), location2,

Log-likelihood: -265.0927 on 196 degrees of freedom

Number of Fisher scoring iterations: 14

No Hauck-Donner effect found in any of the estimates
```

VGAM documentation built on Jan. 16, 2021, 5:21 p.m.