bilogistic: Bivariate Logistic Distribution Family Function

In VGAM: Vector Generalized Linear and Additive Models

Bivariate Logistic Distribution Family Function

Description

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.

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. and 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

## 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)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.