families: Families for Aster Models In aster: 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

 ```1 2 3 4 5 6 7 8``` ```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 = 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 + (k + 1) / (1 + beta)

and the second derivative of the cumulant function has the value

mu [ 1 - (k + 1) / (1 + beta) ( 1 - (k + 1) / mu * beta / (1 + beta) ) ]

.

`"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 = 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 (1 - p) / p

and the second derivative has the value

alpha (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 = 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 + (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.

`aster` and `mlogl`
 ```1 2 3 4 5 6 7 8``` ```### 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)) ```