Bivariate Logistic Distribution Family Function

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

Usage

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 $l_1$ and $l_2$. See Links for more choices.

lscale

Parameter link function applied to both (positive) scale parameters $s_1$ and $s_2$. 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 $$f(y_1,y_2;l_1,s_1,l_2,s_2) = 2 \frac{\exp[-(y_1-l_1)/s_1 - (y_2-l_2)/s_2]}{ s_1 s_2 \left( 1 + \exp[-(y_1-l_1)/s_1] + \exp[-(y_2-l_2)/s_2] \right)^3}$$ where $s_1>0$ and $s_2>0$ are the scale parameters, and $l_1$ and $l_2$ are the location parameters. Each of the two responses are unbounded, i.e., $-\infty<y_j<\infty$. The mean of $Y_1$ is $l_1$ etc. The fitted values are returned in a 2-column matrix. The cumulative distribution function is $$F(y_1,y_2;l_1,s_1,l_2,s_2) = \left( 1 + \exp[-(y_1-l_1)/s_1] + \exp[-(y_2-l_2)/s_2] \right)^{-1}$$ The marginal distribution of $Y_1$ is $$P(Y_1 \leq y_1) = F(y_1;l_1,s_1) = \left( 1 + \exp[-(y_1-l_1)/s_1] \right)^{-1} .$$

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.

Value

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

References

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

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

Author

T. W. Yee

Examples

if (FALSE) { # \dontrun{
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)
} # }