Density, distribution function, quantile function and random generation for the generally--altered, --inflated, --truncated and --deflated Poisson distribution. Both parametric and nonparametric variants are supported; these are based on finite mixtures of the parent with itself and the multinomial logit model (MLM) respectively. Altogether it can be abbreviated as GAAIITDD--Pois(lambda.p)--Pois(lambda.a)--MLM--Pois(lambda.i)--MLM--Pois(lambda.d)--MLM.

```
dgaitdpois(x, lambda.p, a.mix = NULL, a.mlm = NULL, i.mix = NULL,
i.mlm = NULL, d.mix = NULL, d.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0, pstr.mix = 0,
pstr.mlm = 0, pdip.mix = 0, pdip.mlm = 0, byrow.aid = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p,
lambda.d = lambda.p, log = FALSE)
pgaitdpois(q, lambda.p, a.mix = NULL, a.mlm = NULL, i.mix = NULL,
i.mlm = NULL, d.mix = NULL, d.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0, pstr.mix = 0,
pstr.mlm = 0, pdip.mix = 0, pdip.mlm = 0, byrow.aid = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p,
lambda.d = lambda.p, lower.tail = TRUE)
qgaitdpois(p, lambda.p, a.mix = NULL, a.mlm = NULL, i.mix = NULL,
i.mlm = NULL, d.mix = NULL, d.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0, pstr.mix = 0,
pstr.mlm = 0, pdip.mix = 0, pdip.mlm = 0, byrow.aid = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p, lambda.d = lambda.p)
rgaitdpois(n, lambda.p, a.mix = NULL, a.mlm = NULL, i.mix = NULL,
i.mlm = NULL, d.mix = NULL, d.mlm = NULL, truncate = NULL,
max.support = Inf, pobs.mix = 0, pobs.mlm = 0, pstr.mix = 0,
pstr.mlm = 0, pdip.mix = 0, pdip.mlm = 0, byrow.aid = FALSE,
lambda.a = lambda.p, lambda.i = lambda.p, lambda.d = lambda.p)
```

x, q, p, n

Same meaning as in `Poisson`

.

log, lower.tail

Same meaning as in `Poisson`

.

lambda.p, lambda.a, lambda.i, lambda.d

Same meaning as in `Poisson`

,
i.e., for an ordinary Poisson distribution.
The first is for the main *p*arent (or base) distribution.
The next two concern the parametric variant and
these distributions (usually spikes) may be
*a*ltered and/or *i*nflated.
The last one concerns the *d*eflated variant.
Short vectors are recycled.

truncate, max.support

numeric; these specify the set of truncated values.
The default value of `NULL`

means an empty set for the former.
The latter is the
maximum support value so that any value larger
has been truncated (necessary because
`truncate = (max.support + 1):Inf`

is not allowed),
hence is needed for truncating the upper tail of the distribution.
Note that `max(truncate) < max.support`

must be satisfied
otherwise an error message will be issued.

a.mix, i.mix, d.mix

Vectors of nonnegative integers;
the altered, inflated and deflated values for the
parametric variant.
Each argument must have unique values only.
Assigning argument `a.mix`

means that `pobs.mix`

will be used.
Assigning `i.mix`

means that `pstr.mix`

will be used.
Assigning `d.mix`

means that `pdip.mix`

will be used.
If `a.mix`

is of unit length
then the default probability mass function (PMF)
evaluated at `a.mix`

will be `pobs.mix`

.
So having `a.mix = 0`

corresponds to the
zero-inflated Poisson distribution (see `Zipois`

).

a.mlm, i.mlm, d.mlm

Similar to the above, but for the nonparametric (MLM) variant.
For example, assigning `a.mlm`

means that `pobs.mlm`

will be used.
Collectively, the above 7 arguments represent
7 disjoint sets of
special values and they are a proper subset of the support
of the distribution.

pobs.mlm, pstr.mlm, pdip.mlm, byrow.aid

The first three arguments are coerced into a matrix of
probabilities
using `byrow.aid`

to determine the order of the elements
(similar to `byrow`

in `matrix`

, and
the `.aid`

reinforces the behaviour that it applies to both
altered, inflated and deflated cases).
The first argument is recycled if necessary to become
`n x length(a.mlm)`

.
The second argument becomes
`n x length(i.mlm)`

.
The third argument becomes
`n x length(d.mlm)`

.
Thus these arguments are not used unless
`a.mlm`

, `i.mlm`

and `d.mlm`

are assigned.
For deflated models, `pdip.mix`

and `pdip.mlm`

are
positive-valued and VGAM will subtract these quantities;
the argument `deflation`

has been deprecated.

pobs.mix, pstr.mix, pdip.mix

Vectors of probabilities that are recycled if necessary to
length \(n\).
The first argument is used when `a.mix`

is not `NULL`

.
The second argument is used when `i.mix`

is not `NULL`

.
The third argument is used when `d.mix`

is not `NULL`

.

`dgaitdpois`

gives the density,
`pgaitdpois`

gives the distribution function,
`qgaitdpois`

gives the quantile function, and
`rgaitdpois`

generates random deviates.
The default values of the arguments correspond to ordinary
`dpois`

,
`ppois`

,
`qpois`

,
`rpois`

respectively.

These functions allow any combination of 4 operator types:
truncation, alteration, inflation and deflation.
The precedence is
truncation, then alteration and lastly inflation and deflation.
Informally, deflation can be thought of as the opposite of inflation.
This order minimizes the potential interference among the operators.
Loosely, a set of probabilities is set to 0 by truncation
and the remaining probabilities are scaled up.
Then a different set of probabilities are set to some
values `pobs.mix`

and/or `pobs.mlm`

and the remaining probabilities are rescaled up.
Then another different set of probabilities is inflated by
an amount `pstr.mlm`

and/or proportional
to `pstr.mix`

so that individual elements in this set have two sources.
Then another different set of probabilities is deflated by
an amount `pdip.mlm`

and/or proportional
to `pdip.mix`

.
Then all the probabilities are
rescaled so that they sum to unity.

Both parametric and nonparametric variants are implemented.
They usually have arguments with suffix
`.mix`

and `.mlm`

respectively.
The MLM is a loose coupling that effectively separates
the *parent* (or *base*) distribution from
the altered values.
Values inflated nonparametrically effectively have
their spikes shaved off.
The `.mix`

variant has associated with it
`lambda.a`

and `lambda.i`

and `lambda.d`

because it is mixture of 4 Poisson distributions with
partitioned or nested support.

Any value of the support of the distribution that is
altered, inflated, truncated or deflated
is called a *special* value.
A special value that is altered may mean that its probability
increases or decreases relative to the parent distribution.
An inflated special value means that its probability has
increased, provided alteration elsewhere has not made it decrease
in the first case.
There are seven types of special values and they are represented by
`a.mix`

,
`a.mlm`

,
`i.mix`

,
`i.mlm`

,
`d.mix`

,
`d.mlm`

,
`truncate`

.

Terminology-wise, *special* values
are altered or inflated or truncated or deflated, and
the remaining support points that correspond directly to
the parent distribution are *nonspecial* or ordinary.
These functions do what
`Zapois`

,
`Zipois`

,
`Pospois`

collectively did plus much more.

In the notation of Yee and Ma (2022)
these functions allow for the special cases:
(i) GAIT--Pois(`lambda.p`

)--Pois(`lambda.a`

,
`a.mix`

, `pobs.mix`

)--Pois(`lambda.i`

,
`i.mix`

, `pstr.mix`

);
(ii) GAIT--Pois(`lambda.p`

)--MLM(`a.mlm`

,
`pobs.mlm`

)--MLM(`i.mlm`

, `pstr.mlm`

).
Model (i) is totally parametric while model (ii) is the most
nonparametric possible.

Yee, T. W. and Ma, C. (2022).
Generally--altered, --inflated, --truncated and --deflated
regression, with application to heaped and seeped data.
*In preparation*.

`gaitdpoisson`

,
`multinomial`

,
`specials`

,
`spikeplot`

,
`dgaitdplot`

,
`Zapois`

,
`Zipois`

,
`Pospois`

`Poisson`

;
`Gaitdbinom`

,
`Gaitdnbinom`

,
`Gaitdlog`

,
`Gaitdzeta`

.

```
# NOT RUN {
ivec <- c(6, 14); avec <- c(8, 11); lambda <- 10; xgrid <- 0:25
tvec <- 15; max.support <- 20; pobs.mix <- 0.05; pstr.i <- 0.25
dvec <- 13; pdip.mlm <- 0.05; pobs.mlm <- 0.05
(ddd <- dgaitdpois(xgrid, lambda, lambda.a = lambda + 5,
truncate = tvec, max.support = max.support, pobs.mix = pobs.mix,
pobs.mlm = pobs.mlm, a.mlm = avec,
pdip.mlm = pdip.mlm, d.mlm = dvec,
pstr.mix = pstr.i, i.mix = ivec))
# }
# NOT RUN {
dgaitdplot(lambda, ylab = "Probability", xlab = "x",
truncate = tvec, max.support = max.support, pobs.mix = pobs.mix,
pobs.mlm = pobs.mlm, a.mlm = avec, all.lwd = 3,
pdip.mlm = pdip.mlm, d.mlm = dvec,
pstr.mix = pstr.i, i.mix = ivec, deflation = TRUE,
main = "GAITD Combo PMF---Poisson Parent")
# }
```

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