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.

```
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, checkd = FALSE)
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)
```

`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

respectively.

- 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`

.- checkd
Logical. If

`TRUE`

then the density is computed at`floor(q)`

with the same parameters. This can help detect whether the PMF is invalid. If so, then`NaN`

s are returned. See Example 2 below.

It is possible that the GAITD PMF is invalid because
of too much inflation and/or deflation.
This would result in some probabilities exceeding
unity or being negative.
Hence `x`

should ideally contain these types
of special values so that this can be detected.
If so then a `NaN`

is returned and
a warning is issued, e.g.,
same as `dnorm(0, 0, sd = -1)`

.
To help checking,
`pgaitdpois(q, ...)`

calls
`dgaitdpois(floor(q), ...)`

if `checkd`

is `TRUE`

.

That is, given the parameters, it is impractical to determine whether the PMF is valid. To do this, one would have to compute the PMF at all values of its support and check that they are nonnegative and sum to unity. Hence one must be careful to input values from the parameter space, especially for inflation and deflation. See Example 2 below.

T. W. Yee.

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 (2023)
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. (2023)
Generally altered, inflated, truncated and deflated regression.
*In preparation*.

`gaitdpoisson`

,
`multinomial`

,
`specials`

,
`spikeplot`

,
`dgaitdplot`

,
`Zapois`

,
`Zipois`

,
`Pospois`

`Poisson`

;
`Gaitdbinom`

,
`Gaitdnbinom`

,
`Gaitdlog`

,
`Gaitdzeta`

.

```
# Example 1
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))
if (FALSE) 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")
# Example 2: detection of an invalid PMF
xgrid <- 1:3 # Does not cover the special values purposely
(ddd <- dgaitdpois(xgrid, 1, pdip.mlm = 0.1, d.mlm = 5,
pstr.mix = 0.95, i.mix = 0)) # Undetected
xgrid <- 0:13 # Wider range so this detects the problem
(ddd <- dgaitdpois(xgrid, 1, pdip.mlm = 0.1, d.mlm = 5,
pstr.mix = 0.95, i.mix = 0)) # Detected
sum(ddd, na.rm = TRUE) # Something gone awry
```

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