VGAM (version 0.7-10)

bilogistic4: Bivariate Logistic Distribution Family Function

Description

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

Usage

bilogistic4(llocation="identity", lscale="loge",
            iloc1=NULL, iscale1=NULL, iloc2=NULL, iscale2=NULL,
            method.init=1, zero=NULL)

Value

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

Details

The four-parameter bivariate logistic distribution has a density that can be written as $$f(y_1,y_2;l_1,s_1,l_2,s_2) = 2 \frac{\exp[-(y_1-l_1)/s_1 - (y_2-l_2)/s_2]}{ s_1 s_2 \left( 1 + \exp[-(y_1-l_1)/s_1] + \exp[-(y_2-l_2)/s_2] \right)^3}$$ where $s_1>0$ $s_2>0$ are the scale parameters, and $l_1$ and $l_2$ are the location parameters. Each of the two responses are unbounded, i.e., $-\infty

By default, $\eta_1=l_1$, $\eta_2=\log(s_1)$, $\eta_3=l_2$, $\eta_4=\log(s_2)$ are the linear/additive predictors.

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, N.J.: Wiley-Interscience.

See Also

logistic, rbilogis4.

Examples

ymat = rbilogis4(n <- 1000, loc1=5, loc2=7, scale2=exp(1))
plot(ymat)
fit = vglm(ymat ~ 1, fam=bilogistic4, trace=TRUE)
coef(fit, matrix=TRUE)
Coef(fit)
head(fitted(fit))
vcov(fit)
head(weights(fit, type="w"))
summary(fit)