invgamma2mr: 2 - parameter Inverse Gamma Distribution

In VGAMextra: Additions and Extensions of the 'VGAM' Package

2 - parameter Inverse Gamma Distribution

Description

Estimates the 2-parameter Inverse Gamma distribution by maximum likelihood estimation.

Usage

  invgamma2mr(lmu      = "loglink", 
              lshape   = logofflink(offset = -2), 
              parallel = FALSE, 
              ishape   = NULL, 
              imethod  = 1, 
              zero     = "shape") 

Arguments

lmu, lshape	Link functions applied to the (positives) mu and shape parameters (called \mu and a respectively), according to gamma2. See CommonVGAMffArguments for further information.
parallel	Same as gamma2. Details at CommonVGAMffArguments.
ishape	Optional initial value for shape, same as gamma2
imethod	Same as gamma2.
zero	Numeric or character vector. Position or name(s) of the parameters/linear predictors to be modeled as intercept–only. Default is "shape". Details at CommonVGAMffArguments.

Details

The Gamma distribution and the Inverse Gamma distribution are related as follows:Let X be a random variable distributed as Gamma (a, \beta), where a > 0 denotes the shape parameter and \beta > 0 is the scale paramater. Then Y = 1/X is an Inverse Gamma random variable with parameters scale = a and shape = 1/\beta.

The Inverse Gamma density function is given by

f(y;\mu, a) = \frac{(a - 1)^{a} \mu^{a}}{\Gamma(a)}y^{-a- 1} \ e^{-\mu(a - 1)/y},

for \mu > 0, a > 0 and y > 0. Here, \Gamma(\cdot) is the gamma function, as in gamma. The mean of Y is \mu=\mu (returned as the fitted values) with variance \sigma^2 = \mu^2 / (a - 2) if a > 2, else is infinite. Thus, the link function for the shape parameter is logloglink. Then, by default, the two linear/additive predictors are \eta_1=\log(\mu), and \eta_2=\log(a), i.e in the VGLM context, \eta = (log(\mu), loglog(a)

This VGAM family function handles multiple reponses by implementing Fisher scoring and unlike gamma2, the working-weight matrices are not diagonal. The Inverse Gamma distribution is right-skewed and either for small values of a (plus modest \mu) or very large values of \mu (plus moderate a > 2), the density has values too close to zero.

Value

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

Warning

Note that zero can be a numeric or a character vector specifying the position of the names (partially or not) of the linear predictor modeled as intercept only. In this family function such names are

c("mu", "shape").

Numeric values can be entered as usual. See CommonVGAMffArguments for further details.

Note

The response must be strictly positive.

If mu and shape are vectors, then rinvgamma(n = n, shape = shape, scale = mu/(shape - 1) will generate random inverse gamma variates of this parameterization, etc.; see invgammaDist.

Given the math relation between the Gamma and the Inverse Gamma distributions, the parameterization of this VGAM family function underlies on the parametrization of the 2-parameter gamma distribution described in the monograph

Author(s)

Victor Miranda and T. W. Yee

References

McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London, UK. Chapman & Hall.

See Also

invgammaDist, gamma2 for the 2-parameter gamma distribution, GammaDist, CommonVGAMffArguments,

Examples

#------------------------------------------------------------------------#
# Essentially fitting a 2-parameter inverse gamma distribution
# with 2 responses.

set.seed(101)
y1 = rinvgamma(n = 500, scale = exp(2.0), shape = exp(2.0))
y2 = rinvgamma(n = 500, scale = exp(2.5), shape = exp(2.5))
gdata <- data.frame(y1, y2)

fit1 <- vglm(cbind(y1, y2) ~ 1, 
            family = invgamma2mr(zero = NULL, 
            
                                 # OPTIONAL INITIAL VALUE
                                 # ishape = exp(2),
                                 
                                 imethod = 1),
            data = gdata, trace = TRUE)

Coef(fit1)
c(Coef(fit1), log(mean(gdata$y1)), log(mean(gdata$y2)))
summary(fit1)
vcov(fit1, untransform = TRUE)

#------------------------------------------------------------------------#
# An example including one covariate.
# Note that the x2 affects the shape parameter, which implies that both,
# 'mu' and 'shape' are affected.
# Consequently, zero must be set as NULL !

x2    <- runif(1000)
gdata <- data.frame(y3 = rinvgamma(n = 1000, 
                                   scale = exp(2.0), 
                                   shape = exp(2.0 + x2)))

fit2 <- vglm(y3 ~ x2, 
            family = invgamma2mr(lshape = "loglink", zero = NULL), 
            data = gdata, trace = TRUE)

coef(fit2, matrix = TRUE)
summary(fit2)
vcov(fit2)

VGAMextra documentation built on Nov. 2, 2023, 5:59 p.m.