`family`

objects: exponential family of distirbutionsThe **enrichwith** R package provides the `enrich`

method to enrich list-like R objects with new, relevant components. The resulting objects preserve their class, so all methods associated with them still apply.

This vignette is a demo of the available enrichment options for `family`

objects, focusing on objects that correspond to members of the exponential family of distributions.

`family`

objects`family`

objects specify characteristics of the models used by functions such as `glm`

. The families implemented in the `stats`

package include `binomial`

, `gaussian`

, `Gamma`

, `inverse.gaussian`

, and `poisson`

, which obvious corresponding distributions. These distributions are all special cases of the exponential family of distributions with probability mass or density function of the form \[
f_{Y}(y) = \exp\left\{\frac{y \theta - b(\theta) - c_1(y)}{\phi/m} - \frac{1}{2}a\left(-\frac{m}{\phi}\right) + c_2(y) \right\}
\] for some sufficiently smooth functions \(b(.)\), \(c_1(.)\), \(a(.)\) and \(c_2(.)\), and a fixed weight \(m\). The expected value and the variance of \(Y\) are then
\[\begin{align*}
E(Y) & = \mu = b'(\theta) \\
Var(Y) & = \frac{\phi}{m}b''(\theta) = \frac{\phi}{m}V(\mu)
\end{align*}\]
where \(V(\mu)\) and \(\phi\) are the variance function and the dispersion parameter, respectively. Below we list the characteristics of the distributions supported by `family`

objects.

\(\theta = \mu\), \(\displaystyle b(\theta) = \frac{\theta^2}{2}\), \(\displaystyle c_1(y) = \frac{y^2}{2}\), \(\displaystyle a(\zeta) = -\log(-\zeta)\), \(\displaystyle c_2(y) = -\frac{1}{2}\log(2\pi)\)

\(\displaystyle \theta = \log\frac{\mu}{1- \mu}\), \(\displaystyle b(\theta) = \log(1 + e^\theta)\), \(\displaystyle \phi = 1\), \(\displaystyle c_1(y) = 0\), \(\displaystyle a(\zeta) = 0\), \(\displaystyle c_2(y) = \log{m\choose{my}}\)

\(\displaystyle \theta = \log\mu\), \(\displaystyle b(\theta) = e^\theta\), \(\displaystyle \phi = 1\), \(\displaystyle c_1(y) = 0\), \(\displaystyle a(\zeta) = 0\), \(\displaystyle c_2(y) = -\log\Gamma(y + 1)\)

\(\displaystyle \theta = -\frac{1}{\mu}\), \(\displaystyle b(\theta) = -\log(-\theta)\), \(\displaystyle c_1(y) = -\log y\), \(\displaystyle a(\zeta) = 2 \log \Gamma(-\zeta) + 2 \zeta \log\left(-\zeta\right)\), \(\displaystyle c_2(y) = -\log y\)

\(\displaystyle \theta = -\frac{1}{2\mu^2}\), \(\displaystyle b(\theta) = -\sqrt{-2\theta}\), \(\displaystyle c_1(y) = \frac{1}{2y}\), \(\displaystyle a(\zeta) = -\log(-\zeta)\), \(\displaystyle c_2(y) = -\frac{1}{2}\log\left(\pi y^3\right)\)

`family`

objects`family`

objects provide functions for the variance function (`variance`

), a speficiation of deviance residuals (`dev.resids`

) and the Akaike information criterion (`aic`

). For example

`inverse.gaussian()$dev.resids`

```
## function (y, mu, wt)
## wt * ((y - mu)^2)/(y * mu^2)
## <bytecode: 0x7fcb514ce6d8>
## <environment: 0x7fcb514d0b58>
```

`inverse.gaussian()$variance`

```
## function (mu)
## mu^3
## <bytecode: 0x7fcb514ce898>
## <environment: 0x7fcb5151aa40>
```

`inverse.gaussian()$aic`

```
## function (y, n, mu, wt, dev)
## sum(wt) * (log(dev/sum(wt) * 2 * pi) + 1) + 3 * sum(log(y) *
## wt) + 2
## <bytecode: 0x7fcb514cf7b8>
## <environment: 0x7fcb5157d908>
```

`family`

objectsThe **enrichwith** R package provides methods for the enrichment of `family`

objects with a function that links the natural parameter \(\theta\) with \(\mu\), the function \(b(\theta)\), the first two derivatives of \(V(\mu)\), \(a(\zeta)\) and its first four derivatives, and \(c_1(y)\) and \(c_2(y)\). To illustrate, letâ€™s write a function that reconstructs the densitities and probability mass functions from the components that result from enrichment

```
dens <- function(y, m = 1, mu, phi, family) {
object <- enrich(family)
with(object, {
c2 <- if (family == "binomial") c2fun(y, m) else c2fun(y)
exp(m * (y * theta(mu) - bfun(theta(mu)) - c1fun(y))/phi -
0.5 * afun(-m/phi) + c2)
})
}
```

The following chunks test `dens`

for a few distributions against the standard `d*`

functions

```
## Normal
all.equal(dens(y = 0.2, m = 3, mu = 1, phi = 3.22, gaussian()),
dnorm(x = 0.2, mean = 1, sd = sqrt(3.22/3)))
```

`## [1] TRUE`

```
## Gamma
all.equal(dens(y = 3, m = 1.44, mu = 2.3, phi = 1.3, Gamma()),
dgamma(x = 3, shape = 1.44/1.3, 1.44/(1.3 * 2.3)))
```

`## [1] TRUE`

```
## Inverse gaussian
all.equal(dens(y = 0.2, m = 7.23, mu = 1, phi = 3.22, inverse.gaussian()),
SuppDists::dinvGauss(0.2, nu = 1, lambda = 7.23/3.22))
```

`## [1] TRUE`

```
## Binomial
all.equal(dens(y = 0.34, m = 100, mu = 0.32, phi = 1, binomial()),
dbinom(x = 34, size = 100, prob = 0.32))
```

`## [1] TRUE`