families: Families for Aster Models

In aster2: Aster Models

Families for Aster Models

Description

Families known to the package. These functions construct simple family specifications used in specifying aster models. Statistical properties of these families are described.

Usage

fam.bernoulli()
fam.poisson()
fam.zero.truncated.poisson()
fam.normal.location.scale()
fam.multinomial(dimension)

Arguments

dimension

the dimension (number of categories) for the multinomial distribution.

Details

Currently implemented families are

"bernoulli"

Bernoulli (binomial with sample size one). The distribution of any zero-or-one-valued random variable Y, which is the canonical statistic. The mean value parameter is

\mu = E(Y) = \Pr(Y = 1).

The canonical parameter is \theta = \log(\mu) - \log(1 - \mu), also called logit of \mu. The cumulant function is

c(\theta) = \log(1 + e^\theta).

This distribution has degenerate limiting distributions. The lower limit as \theta \to - \infty is the distribution concentrated at zero, having cumulant function which is the constant function everywhere equal to zero. The upper limit as \theta \to + \infty is the distribution concentrated at one, having cumulant function which is the identity function satisfying c(\theta) = \theta for all \theta.

For predecessor (sample size) n, the successor is the sum of n independent and identically distributed (IID) Bernoulli random variables, that is, binomial with sample size n. The mean value parameter is n times the mean value parameter for sample size one; the cumulant function is n times the cumulant function for sample size one; the canonical parameter is the same for all sample sizes.

"poisson"

Poisson. The mean value parameter \mu is the mean of the Poisson distribution. The canonical parameter is \theta = \log(\mu). The cumulant function is

c(\theta) = e^\theta.

This distribution has a degenerate limiting distribution. The lower limit as \theta \to - \infty is the distribution concentrated at zero, having cumulant function which is the constant function everywhere equal to zero. There is no upper limit because the canonical statistic is unbounded above.

For predecessor (sample size) n, the successor is the sum of n IID Poisson random variables, that is, Poisson with mean n \mu. The mean value parameter is n times the mean value parameter for sample size one; the cumulant function is n times the cumulant function for sample size one; the canonical parameter is the same for all sample sizes.

"zero.truncated.poisson"

Poisson conditioned on being greater than zero. Let m be the mean of the corresponding untruncated Poisson distribution. Then the canonical parameters for both truncated and untruncated distributions are the same \theta = \log(m). The mean value parameter for the zero-truncated Poisson distribution is

\mu = \frac{m}{1 - e^{- m}}

and the cumulant function is

c(\theta) = m + \log(1 - e^{- m}),

where m is as defined above, so m = e^\theta.

This distribution has a degenerate limiting distribution. The lower limit as \theta \to - \infty is the distribution concentrated at one, having cumulant function which is the identity function satisfying c(\theta) = \theta for all \theta. There is no upper limit because the canonical statistic is unbounded above.

For predecessor (sample size) n, the successor is the sum of n IID zero-truncated Poisson random variables, which is not a brand-name distribution. The mean value parameter is n times the mean value parameter for sample size one; the cumulant function is n times the cumulant function for sample size one; the canonical parameter is the same for all sample sizes.

"normal.location.scale"

The distribution of a normal random variable X with unknown mean m and unknown variance v. Thought of as an exponential family, this is a two-parameter family, hence must have a two-dimensional canonical statistic Y = (X, X^2). The canonical parameter vector \theta has components

\theta_1 = \frac{m}{v}

and

\theta_2 = - \frac{1}{2 v}.

The value of \theta_1 is unrestricted, but \theta_2 must be strictly negative. The mean value parameter vector \mu has components

\mu_1 = m = - \frac{\theta_1}{2 \theta_2}

and

\mu_2 = v + m^2 = - \frac{1}{2 \theta_2} + \frac{\theta_1^2}{4 \theta_2^2}.

The cumulant function is

c(\theta) = - \frac{\theta_1^2}{4 \theta_2} + \frac{1}{2} \log\left(- \frac{1}{2 \theta_2}\right).

This distribution has no degenerate limiting distributions, because the canonical statistic is a continuous random vector so the boundary of its support has probability zero.

For predecessor (sample size) n, the successor is the sum of n IID random vectors (X_i, X_i^2), where each X_i is normal with mean m and variance v, and this is not a brand-name multivariate distribution (the first component of the sum is normal, the second component noncentral chi-square, and the components are not independent). The mean value parameter vector is n times the mean value parameter vector for sample size one; the cumulant function is n times the cumulant function for sample size one; the canonical parameter vector is the same for all sample sizes.

"multinomial"

Multinomial with sample size one. The distribution of any random vector Y having all components zero except for one component which is one (Y is the canonical statistic vector). The mean value parameter is the vector \mu = E(Y) having components

\mu_i = E(Y_i) = \Pr(Y_i = 1).

The mean value parameter vector \mu is given as a function of the canonical parameter vector \theta by

\mu_i = \frac{e^{\theta_i}}{\sum_{j = 1}^d e^{\theta_j}},

where d is the dimension of Y and \theta and \mu. This transformation is not one-to-one; adding the same number to each component of \theta does not change the value of \mu. The cumulant function is

c(\theta) = \log\left(\sum_{j = 1}^d e^{\theta_j}\right).

This distribution is degenerate. The sum of the components of the canonical statistic is equal to one with probability one, which implies the nonidentifiability of the d-dimensional canonical parameter vector mentioned above. Hence one parameter (at least) is always constrained to to be zero in fitting an aster model with a multinomial family.

This distribution has many degenerate distributions. For any vector \delta the limit of distributions having canonical parameter vectors \theta + s \delta as s \to \infty exists and is another multinomial distribution (the limit distribution in the direction \delta). Let A be the set of i such that \delta_i = \max(\delta), where \max(\delta) denotes the maximum over the components of \delta. Then the limit distribution in the direction \delta has components Y_i of the canonical statistic for i \notin A concentrated at zero. The cumulant function of this degenerate distribution is

c(\theta) = \log\left(\sum_{j \in A} e^{\theta_j}\right).

The canonical parameters \theta_j for j \notin A are not identifiable, and one other canonical parameter is not identifiable because of the constraint that the sum of the components of the canonical statistic is equal to one with probability one.

For predecessor (sample size) n, the successor is the sum of n IID multinomial-sample-size-one random vectors, that is, multinomial with sample size n. The mean value parameter is n times the mean value parameter for sample size one; the cumulant function is n times the cumulant function for sample size one; the canonical parameter is the same for all sample sizes.

Value

a list of class "astfam" giving name and values of any hyperparameters.

Examples

fam.bernoulli()
fam.multinomial(4)

aster2 documentation built on Sept. 18, 2024, 1:06 a.m.