ZeroTruncatedPoisson {actuar} R Documentation

## The Zero-Truncated Poisson Distribution

### Description

Density function, distribution function, quantile function, random generation for the Zero-Truncated Poisson distribution with parameter `lambda`.

### Usage

```dztpois(x, lambda, log = FALSE)
pztpois(q, lambda, lower.tail = TRUE, log.p = FALSE)
qztpois(p, lambda, lower.tail = TRUE, log.p = FALSE)
rztpois(n, lambda)
```

### Arguments

 `x` vector of (strictly positive integer) quantiles. `q` vector of quantiles. `p` vector of probabilities. `n` number of values to return. `lambda` vector of (non negative) means. `log, log.p` logical; if `TRUE`, probabilities p are returned as log(p). `lower.tail` logical; if `TRUE` (default), probabilities are P[X <= x], otherwise, P[X > x].

### Details

The zero-truncated Poisson distribution has probability mass function

p(x) = lambda^x exp(-lambda)/[x! (1 - exp(-lambda))] = lambda^x/[x! (e^lambda - 1)]

for x = 1, 2, ..., and p(1) = 1 when λ = 0. The cumulative distribution function is

P(x) = [F(x) - F(0)]/[1 - F(0)],

where F(x) is the distribution function of the standard Poisson.

The mean is λ/(1 - exp(-λ)) and the variance is λ[1 - (λ+1)exp(-λ)]/(1 - exp(-λ))^2.

In the terminology of Klugman et al. (2012), the zero-truncated Poisson is a member of the (a, b, 1) class of distributions with a = 0 and b = λ.

If an element of `x` is not integer, the result of `dztpois` is zero, with a warning.

The quantile is defined as the smallest value x such that P(x) ≥ p, where P is the distribution function.

### Value

`dztpois` gives the (log) probability mass function, `pztpois` gives the (log) distribution function, `qztpois` gives the quantile function, and `rztpois` generates random deviates.

Invalid `lambda` will result in return value `NaN`, with a warning.

The length of the result is determined by `n` for `rztpois`, and is the maximum of the lengths of the numerical arguments for the other functions.

### Note

Functions `{d,p,q}ztpois` use `{d,p,q}pois` for all but the trivial input values and p(0).

`rztpois` uses the simple inversion algorithm suggested by Peter Dalgaard on the r-help mailing list on 1 May 2005 (https://stat.ethz.ch/pipermail/r-help/2005-May/070680.html).

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca

### References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

### See Also

`dpois` for the standard Poisson distribution.

### Examples

```dztpois(1:5, lambda = 1)
dpois(1:5, lambda = 1)/ppois(0, 1, lower = FALSE) # same

pztpois(1, lambda = 0)          # point mass at 1

qztpois(pztpois(1:10, 1), 1)

x <- seq(0, 8)
plot(x, dztpois(x, 2), type = "h", lwd = 2, ylab = "p(x)",
main = "Zero-Truncated Poisson(2) and Poisson(2) PDF")
points(x, dpois(x, 2), pch = 19, col = "red")
legend("topright", c("ZT Poisson probabilities", "Poisson probabilities"),
col = c("black", "red"), lty = c(1, 0), lwd = 2, pch = c(NA, 19))
```

[Package actuar version 3.1-4 Index]