Poisson-Lindley Distribution

These functions provide density, distribution function, quantile function, and random number generation for the Poisson-Lindley (PL) Distribution

Usage

msg1

dplind(x, mean = 1, theta = 1, lambda = NULL, log = FALSE)

pplind(q, mean = 1, theta = 1, lambda = NULL, lower.tail = TRUE, log.p = FALSE)

qplind(p, mean = 1, theta = 1, lambda = NULL)

rplind(n, mean = 1, theta = 1, lambda = NULL)

Arguments

x

numeric value or a vector of values.

mean

numeric value or vector of mean values for the distribution (the values have to be greater than 0).

theta

single value or vector of values for the theta parameter of the distribution (the values have to be greater than 0).

lambda

alternative parameterization (use instead of the mean); numeric value or vector of values for lambda parameter of the distribution (the values have to be greater than 0).

log

logical; if TRUE, probabilities p are given as log(p).

q

quantile or a vector of quantiles.

lower.tail

logical; if TRUE, probabilities p are \(P[X\leq x]\) otherwise, \(P[X>x]\).

log.p

logical; if TRUE, probabilities p are given as log(p).

p

probability or a vector of probabilities.

n

the number of random numbers to generate.

Value

dplind gives the density, pplind gives the distribution function, qplind gives the quantile function, and rplind generates random deviates.

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

Details

The Poisson-Lindley is a 2-parameter count distribution that captures high densities for small integer values. This makes it ideal for data that are zero-inflated.

dplind computes the density (PDF) of the Poisson-Lindley Distribution.

pplind computes the CDF of the Poisson-Lindley Distribution.

qplind computes the quantile function of the Poisson-Lindley Distribution.

rplind generates random numbers from the Poisson-Lindley Distribution.

The compound Probability Mass Function (PMF) for the Poisson-Lindley (PL) distribution is: $$f(y| \theta, \lambda) = \frac{\theta^2 \lambda^y (\theta + \lambda + y + 1)} {(\theta + 1)(\theta + \lambda)^{y + 2}}$$

Where \(\theta\) and \(\lambda\) are distribution parameters with the restrictions that \(\theta > 0\) and \(\lambda > 0\), and \(y\) is a non-negative integer.

The expected value of the distribution is: $$\mu=\frac{\lambda(\theta + 2)}{\theta(\theta + 1)}$$

The default is to use the input mean value for the distribution. However, the lambda parameter can be used as an alternative to the mean value.

Examples

dplind(0, mean = 0.75, theta = 7)
#> [1] 0.57
pplind(c(0, 1, 2, 3, 5, 7, 9, 10), mean = 0.75, theta = 7)
#> [1] 0.5700000 0.8160000 0.9216000 0.9667200 0.9940608 0.9989514 0.9998165
#> [8] 0.9999235
qplind(c(0.1, 0.3, 0.5, 0.9, 0.95), lambda = 4.67, theta = 7)
#> [1] 0 0 0 2 3
rplind(30, mean = 0.75, theta = 7)
#>  [1] 0 1 0 0 5 1 2 3 1 1 1 2 1 0 3 0 0 1 1 0 1 0 0 0 1 0 0 0 2 1