bilogistic4 {VGAM} | R Documentation |
Estimates the four parameters of the bivariate logistic distribution by maximum likelihood estimation.
bilogistic4(llocation="identity", lscale="loge", iloc1=NULL, iscale1=NULL, iloc2=NULL, iscale2=NULL, method.init=1, zero=NULL)
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
method.init . Assigning values here will override
the argument method.init . |
iscale1, iscale2 |
Initial values for the scale parameters.
By default, initial values are chosen internally using
method.init . Assigning values here will override
the argument method.init . |
method.init |
An integer with value 1 or 2 which
specifies the initialization method. If failure to converge occurs
try the other value. |
zero |
An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The default is none of them. If used, choose values from the set {1,2,3,4}. |
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 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.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
rrvglm
and vgam
.
This family function uses the BFGS quasi-Newton update formula for the
working weight matrices. Consequently the estimated variance-covariance
matrix may be inaccurate or simply wrong! The standard errors must be
therefore treated with caution; these are computed in functions such
as vcov()
and summary()
.
T. W. Yee
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.
ymat = rbilogis4(n <- 1000, loc1=5, loc2=7, scale2=exp(1)) ## Not run: plot(ymat) fit = vglm(ymat ~ 1, fam=bilogistic4, trace=TRUE) coef(fit, matrix=TRUE) Coef(fit) fitted(fit)[1:4,] vcov(fit) weights(fit, type="w")[1:4,] summary(fit)