# geometric

##### Geometric (Truncated and Untruncated) Distributions

Maximum likelihood estimation for the geometric and truncated geometric distributions.

- Keywords
- models, regression

##### Usage

```
geometric(link = "logitlink", expected = TRUE, imethod = 1,
iprob = NULL, zero = NULL)
truncgeometric(upper.limit = Inf,
link = "logitlink", expected = TRUE, imethod = 1,
iprob = NULL, zero = NULL)
```

##### Arguments

- link
Parameter link function applied to the probability parameter \(p\), which lies in the unit interval. See

`Links`

for more choices.- expected
Logical. Fisher scoring is used if

`expected = TRUE`

, else Newton-Raphson.- iprob, imethod, zero
See

`CommonVGAMffArguments`

for details.- upper.limit
Numeric. Upper values. As a vector, it is recycled across responses first. The default value means both family functions should give the same result.

##### Details

A random variable \(Y\) has a 1-parameter geometric distribution
if \(P(Y=y) = p (1-p)^y\)
for \(y=0,1,2,\ldots\).
Here, \(p\) is the probability of success,
and \(Y\) is the number of (independent) trials that are fails
until a success occurs.
Thus the response \(Y\) should be a non-negative integer.
The mean of \(Y\) is \(E(Y) = (1-p)/p\)
and its variance is \(Var(Y) = (1-p)/p^2\).
The geometric distribution is a special case of the
negative binomial distribution (see `negbinomial`

).
The geometric distribution is also a special case of the
Borel distribution, which is a Lagrangian distribution.
If \(Y\) has a geometric distribution with parameter \(p\) then
\(Y+1\) has a positive-geometric distribution with the same parameter.
Multiple responses are permitted.

For `truncgeometric()`

,
the (upper) truncated geometric distribution can have response integer
values from 0 to `upper.limit`

.
It has density `prob * (1 - prob)^y / [1-(1-prob)^(1+upper.limit)]`

.

For a generalized truncated geometric distribution with integer values \(L\) to \(U\), say, subtract \(L\) from the response and feed in \(U-L\) as the upper limit.

##### Value

An object of class `"vglmff"`

(see `vglmff-class`

).
The object is used by modelling functions such as `vglm`

,
and `vgam`

.

##### References

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011)
*Statistical Distributions*,
Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

##### See Also

`negbinomial`

,
`Geometric`

,
`betageometric`

,
`expgeometric`

,
`zageometric`

,
`zigeometric`

,
`rbetageom`

,
`simulate.vlm`

.

##### Examples

```
# NOT RUN {
gdata <- data.frame(x2 = runif(nn <- 1000) - 0.5)
gdata <- transform(gdata, x3 = runif(nn) - 0.5,
x4 = runif(nn) - 0.5)
gdata <- transform(gdata, eta = -1.0 - 1.0 * x2 + 2.0 * x3)
gdata <- transform(gdata, prob = logitlink(eta, inverse = TRUE))
gdata <- transform(gdata, y1 = rgeom(nn, prob))
with(gdata, table(y1))
fit1 <- vglm(y1 ~ x2 + x3 + x4, geometric, data = gdata, trace = TRUE)
coef(fit1, matrix = TRUE)
summary(fit1)
# Truncated geometric (between 0 and upper.limit)
upper.limit <- 5
tdata <- subset(gdata, y1 <= upper.limit)
nrow(tdata) # Less than nn
fit2 <- vglm(y1 ~ x2 + x3 + x4, truncgeometric(upper.limit),
data = tdata, trace = TRUE)
coef(fit2, matrix = TRUE)
# Generalized truncated geometric (between lower.limit and upper.limit)
lower.limit <- 1
upper.limit <- 8
gtdata <- subset(gdata, lower.limit <= y1 & y1 <= upper.limit)
with(gtdata, table(y1))
nrow(gtdata) # Less than nn
fit3 <- vglm(y1 - lower.limit ~ x2 + x3 + x4,
truncgeometric(upper.limit - lower.limit),
data = gtdata, trace = TRUE)
coef(fit3, matrix = TRUE)
# }
```

*Documentation reproduced from package VGAM, version 1.1-1, License: GPL-3*