frank {VGAM} | R Documentation |
Estimate the association parameter of Frank's bivariate distribution by maximum likelihood estimation.
frank(lapar="loge", eapar=list(), iapar=2, nsimEIM=250)
lapar |
Link function applied to the (positive) association parameter
alpha.
See Links for more choices.
|
eapar |
List. Extra argument for the link.
See earg in Links for general information.
|
iapar |
Numeric. Initial value for alpha.
If a convergence failure occurs try assigning a different value.
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nsimEIM |
See CommonVGAMffArguments .
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The cumulative distribution function is
P(Y1 <= y1, Y2 <= y2) = H_{alpha}(y1,y2) = log_{alpha} [1 + (alpha^(y1)-1)*(alpha^(y2)-1)/ (alpha-1)]
for alpha != 1. Note the logarithm here is to base alpha. The support of the function is the unit square.
When 0<alpha<1 the probability density function h_{alpha}(y_1,y_2) is symmetric with respect to the lines y2=y1 and y2=1-y1. When alpha>1 then h_{1/alpha}(1-y_1,y_2).
If alpha=1 then H(y1,y2)=y1*y2, i.e., uniform on the unit square. As alpha approaches 0 then H(y1,y2)=min(y1,y2). As alpha approaches infinity then H(y1,y2)=max(0,y1+y2-1).
The default is to use Fisher scoring implemented using
rfrank
.
For intercept-only models an alternative is to set nsimEIM=NULL
so that a variant of Newton-Raphson is used.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
and vgam
.
The response must be a two-column matrix. Currently, the fitted value is a matrix with two columns and values equal to a half. This is because the marginal distributions correspond to a standard uniform distribution.
T. W. Yee
Genest, C. (1987) Frank's family of bivariate distributions. Biometrika, 74, 549–555.
ymat = rfrank(n=2000, alpha=exp(4)) ## Not run: plot(ymat, col="blue") fit = vglm(ymat ~ 1, fam=frank, trace=TRUE) coef(fit, matrix=TRUE) Coef(fit) vcov(fit) fitted(fit)[1:5,] summary(fit)