families {aster2} | R Documentation |
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
, which is the canonical statistic. The mean value parameter is
The canonical parameter is
, also called logit of
. The cumulant function is
This distribution has degenerate limiting distributions. The lower limit as
is the distribution concentrated at zero, having cumulant function which is the constant function everywhere equal to zero. The upper limit as
is the distribution concentrated at one, having cumulant function which is the identity function satisfying
for all
.
For predecessor (sample size)
, the successor is the sum of
independent and identically distributed (IID) Bernoulli random variables, that is, binomial with sample size
. The mean value parameter is
times the mean value parameter for sample size one; the cumulant function is
times the cumulant function for sample size one; the canonical parameter is the same for all sample sizes.
"poisson"
Poisson. The mean value parameter
is the mean of the Poisson distribution. The canonical parameter is
. The cumulant function is
This distribution has a degenerate limiting distribution. The lower limit as
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)
, the successor is the sum of
IID Poisson random variables, that is, Poisson with mean
. The mean value parameter is
times the mean value parameter for sample size one; the cumulant function is
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
be the mean of the corresponding untruncated Poisson distribution. Then the canonical parameters for both truncated and untruncated distributions are the same
. The mean value parameter for the zero-truncated Poisson distribution is
and the cumulant function is
where
is as defined above, so
.
This distribution has a degenerate limiting distribution. The lower limit as
is the distribution concentrated at one, having cumulant function which is the identity function satisfying
for all
. There is no upper limit because the canonical statistic is unbounded above.
For predecessor (sample size)
, the successor is the sum of
IID zero-truncated Poisson random variables, which is not a brand-name distribution. The mean value parameter is
times the mean value parameter for sample size one; the cumulant function is
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
with unknown mean
and unknown variance
. Thought of as an exponential family, this is a two-parameter family, hence must have a two-dimensional canonical statistic
. The canonical parameter vector
has components
and
The value of
is unrestricted, but
must be strictly negative. The mean value parameter vector
has components
and
The cumulant function is
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)
, the successor is the sum of
IID random vectors
, where each
is normal with mean
and variance
, 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
times the mean value parameter vector for sample size one; the cumulant function is
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
having all components zero except for one component which is one (
is the canonical statistic vector). The mean value parameter is the vector
having components
The mean value parameter vector
is given as a function of the canonical parameter vector
by
where
is the dimension of
and
and
. This transformation is not one-to-one; adding the same number to each component of
does not change the value of
. The cumulant function is
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
-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
the limit of distributions having canonical parameter vectors
as
exists and is another multinomial distribution (the limit distribution in the direction
). Let
be the set of
such that
, where
denotes the maximum over the components of
. Then the limit distribution in the direction
has components
of the canonical statistic for
concentrated at zero. The cumulant function of this degenerate distribution is
The canonical parameters
for
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)
, the successor is the sum of
IID multinomial-sample-size-one random vectors, that is, multinomial with sample size
. The mean value parameter is
times the mean value parameter for sample size one; the cumulant function is
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)