families: Families for Aster Models

In aster: Aster Models

Families for Aster Models

Description

Families (response models) known to the package. These functions construct simple family specifications used in specifying models for aster and mlogl. They are mostly for convenience, since the specifications are easy to construct by hand.

Usage

fam.bernoulli()
fam.poisson()
fam.truncated.poisson(truncation)
fam.negative.binomial(size)
fam.truncated.negative.binomial(size, truncation)
fam.normal.location(sd)
fam.default()
famfun(fam, deriv, theta)

Arguments

truncation	the truncation point, called k in the details section below.
size	the sample size. May be non-integer.
sd	the standard deviation. May be non-integer.
fam	a family specification, which is a list of class "astfam" containing, at least one element named "name" and perhaps other elements specifying hyperparameters.
deriv	derivative wanted: 0, 1, or 2.
theta	value of the canonical parameter.

Details

Currently implemented families are

"bernoulli"

Bernoulli. The mean value parameter \mu is the success probability. The canonical parameter is \theta = \log(\mu) - \log(1 - \mu), also called logit of \mu. The first derivative of the cumulant function has the value \mu and the second derivative of the cumulant function has the value \mu (1 - \mu).

"poisson"

Poisson. The mean value parameter \mu is the mean of the Poisson distribution. The canonical parameter is \theta = \log(\mu). The first and second derivatives of the cumulant function both have the value \mu.

"truncated.poisson"

Poisson conditioned on being strictly greater than k, specified by the argument truncation. Let \mu be the mean of the corresponding untruncated Poisson distribution. Then the canonical parameters for both truncated and untruncated distributions are the same \theta = \log(\mu). Let Y be a Poisson random variable having the same mean parameter as this distribution, and define

\beta = \frac{\Pr\{Y > k + 1\}}{\Pr\{Y = k + 1\}}

Then the mean value parameter and first derivative of the cumulant function of this distribution has the value

\tau = \mu + \frac{k + 1}{1 + \beta}

and the second derivative of the cumulant function has the value

\mu \left[ 1 - \frac{k + 1}{1 + \beta} \left( 1 - \frac{k + 1}{\mu} \cdot \frac{\beta}{1 + \beta} \right) \right]

.

"negative.binomial"

Negative binomial. The size parameter \alpha may be noninteger, meaning the cumulant function is \alpha times the cumulant function of the geometric distribution. The mean value parameter \mu is the mean of the negative binomial distribution. The success probability parameter is

p = \frac{\alpha}{\mu + \alpha}.

The canonical parameter is \theta = \log(1 - p). Since 1 - p < 1, the canonical parameter space is restricted, the set of \theta such that \theta < 0. This is, however, a regular exponential family (the log likelihood goes to minus infinity as \theta converges to the boundary of the parameter space, so the constraint \theta < 0 plays no role in maximum likelihood estimation so long as the optimization software is not too stupid. There will be no problems so long as the default optimizer (trust) is used. Since zero is not in the canonical parameter space a negative default origin is used. The first derivative of the cumulant function has the value

\mu = \alpha \frac{1 - p}{p}

and the second derivative has the value

\alpha \frac{1 - p}{p^2}.

"truncated.negative.binomial"

Negative binomial conditioned on being strictly greater than k, specified by the argument truncation. Let p be the success probability parameter of the corresponding untruncated negative binomial distribution. Then the canonical parameters for both truncated and untruncated distributions are the same \theta = \log(1 - p), and consequently the canonical parameter spaces are the same, the set of \theta such that \theta < 0, and both models are regular exponential families. Let Y be an untruncated negative binomial random variable having the same size and success probability parameters as this distribution. and define

\beta = \frac{\Pr\{Y > k + 1\}}{\Pr\{Y = k + 1\}}

Then the mean value parameter and first derivative of the cumulant function of this distribution has the value

\tau = \mu + \frac{k + 1}{p (1 + \beta)}

and the second derivative is too complicated to write here (the formula can be found in the vignette trunc.pdf.

"normal.location"

Normal, unknown mean, known variance. The sd (standard deviation) parameter \sigma may be noninteger, meaning the cumulant function is \sigma^2 times the cumulant function of the standard normal distribution. The mean value parameter \mu is the mean of the normal distribution. The canonical parameter is \theta = \mu / \sigma^2. The first derivative of the cumulant function has the value

\mu = \sigma^2 \theta

and the second derivative has the value

\sigma^2.

Value

For all but fam.default, a list of class "astfam" giving name and values of any hyperparameters. For fam.default, a list each element of which is of class "astfam". The list of families which were hard coded in earlier versions of the package.

See Also

aster and mlogl

Examples

### mean of poisson with mean 0.2
famfun(fam.poisson(), 1, log(0.2))
### variance of poisson with mean 0.2
famfun(fam.poisson(), 2, log(0.2))
### mean of poisson with mean 0.2 conditioned on being nonzero
famfun(fam.truncated.poisson(trunc = 0), 1, log(0.2))
### variance of poisson with mean 0.2 conditioned on being nonzero
famfun(fam.truncated.poisson(trunc = 0), 2, log(0.2))

aster documentation built on May 29, 2024, 2:09 a.m.