families {aster} | R Documentation |

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.

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)

`truncation` |
the truncation point, called |

`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 |

`deriv` |
derivative wanted: 0, 1, or 2. |

`theta` |
value of the canonical parameter. |

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.*

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.

### 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))

[Package *aster* version 1.1-2 Index]