bilogistic: Bivariate Logistic Distribution Family Function

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/family.bivariate.R

Description

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

Usage

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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.

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

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## 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
Loading required package: stats4
Loading required package: splines
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
            location1 loglink(scale1) location2 loglink(scale2)
(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, 
loglink(scale2)

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.