# linkfcn: Calculate the link function for exponential families In geoBayes: Analysis of Geostatistical Data using Bayes and Empirical Bayes Methods

## Description

Link function for the exponential family.

## Usage

 ```1 2 3``` ```linkfcn(mu, linkp, family = "gaussian") linkinv(z, linkp, family = "gaussian") ```

## Arguments

 `mu` Numeric. The mean of the response variable. `linkp` The link function parameter. A scalar. `family` The distribution of the response variable from `.geoBayes_models`. Either an integer or the family name. `z` Numeric. The linear predictor.

## Details

`linkfcn` maps the mean of the response variable `mu` to the linear predictor `z`. `linkinv` is its inverse.

For the Gaussian family, if the link parameter is positive, then the extended link is used, defined by

z = (sign(mu)*abs(mu)^nu - 1)/nu

In the other case, the usual Box-Cox link is used.

For the Poisson and gamma families, if the link parameter is positive, then the link is defined by

z = sign(w)*expm1(nu*w)/nu

where w = log(mu). In the other case, the usual Box-Cox link is used.

For the GEV binomial family, the link function is defined by

mu = 1 - exp[-max(0, 1 + nu z)^(1/nu)]

for any real nu. At nu = 0 it reduces to the complementary log-log link.

The Wallace binomial family is a fast approximation to the robit family. It is defined as

mu = Phi(sign(z) c(nu) sqrt{nu log(1 + z^2/nu)})

where c(nu) = (8*nu+1)/(8*nu+3)

## Value

A numeric array of the same dimension as the function's first argument.

## References

Evangelou, E., & Roy, V. (2019). Estimation and prediction for spatial generalized linear mixed models with parametric links via reparameterized importance sampling. Spatial Statistics, 29, 289-315.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```## Not run: mu <- seq(0.1, 0.9, 0.1) linkfcn(mu, 7, "binomial") # robit(7) link function linkfcn(mu, , "binomial.logit") # logit link function mu <- seq(-3, 3, 1) linkfcn(mu, 0.5, "gaussian") # sqrt transformation linkinv(linkfcn(mu, 0.5, "gaussian"), 0.5, "gaussian") curve(linkfcn(x, 0.5, "gaussian"), -3, 3) ## End(Not run) ```

geoBayes documentation built on Dec. 2, 2019, 5:13 p.m.