Poisson-Lindley-Gamma (Negative Binomial-Lindley) Distribution

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

Usage

dplindGamma(
  x,
  mean = 1,
  theta = 1,
  alpha = 1,
  log = FALSE,
  rel.tol = 1e-08,
  subdivisions = 200L
)

pplindGamma(
  q,
  mean = 1,
  theta = 1,
  alpha = 1,
  lower.tail = TRUE,
  log.p = FALSE,
  rel.tol = 1e-08,
  subdivisions = 200L
)

qplindGamma(p, mean = 1, theta = 1, alpha = 1)

rplindGamma(n, mean = 1, theta = 1, alpha = 1)

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).

alpha

single value or vector of values for the alpha parameter of the gamma distribution in the special case that the mean = 1 and the variance = alpha (the values for alpha have to be greater than 0).

log

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

rel.tol

Relative numerical integration tolerance; default 1e-8.

subdivisions

Maximum subintervals per component-side integral.

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

dplindGamma gives the density, pplindGamma gives the distribution function, qplindGamma gives the quantile function, and rplindGamma generates random deviates.

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

Details

The Poisson-Lindley-Gamma is a count distribution that captures high densities for small integer values and provides flexibility for heavier tails.

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

pplindGamma computes the CDF of the Poisson-Lindley-Gamma Distribution.

qplindGamma computes the quantile function of the Poisson-Lindley-Gamma Distribution.

rplindGamma generates random numbers from the Poisson-Lindley-Gamma Distribution.

The compound Probability Mass Function (PMF) for the Poisson-Lindley-Gamma (PLG) distribution is: $$ f(x|\mu,\theta,\alpha)= \frac{ (\theta+2)^2\Gamma(x+1/\alpha) }{ \alpha\mu^2(\theta+1)^3\Gamma(1/\alpha) } \left( \frac{\mu\theta(\theta+1)}{\theta+2} U\left( x+1,2-1/\alpha,\frac{1/\alpha(\theta+2)}{\mu(\theta+1)} \right) + 1/\alpha(x+1) U\left( x+2,3-1/\alpha,\frac{1/\alpha(\theta+2)}{\mu(\theta+1)} \right) \right) $$

Where \(\theta\) is a distribution parameter from the Poisson-Lindley distribution with the restrictions that \(\theta>0\), \(\alpha\) is a parameter for the gamma distribution with the restriction \(\alpha>0\), \(\mu\) is the mean value, and \(x\) is a non-negative integer, and $$U(a,b,z)$$ is the Tricomi’s confluent hypergeometric function to - also known as the confluent hypergeometric function of the second kind

The expected value of the distribution is: $$E[x]=\mu$$

The variance is: $$\sigma^2=\mu+\left(\left(1+\alpha\right)\left(2-\frac{2} {(\theta+2)^2}\right)-1\right)\mu^2$$

Examples

dplindGamma(0, mean=0.75, theta=7, alpha=2)
#> [1] 0.6987929
pplindGamma(c(0,1,2,3,5,7,9,10), mean=0.75, theta=3, alpha=0.5)
#> [1] 0.6124292 0.8264579 0.9149954 0.9556459 0.9861974 0.9951259 0.9981156
#> [8] 0.9987963
qplindGamma(c(0.1,0.3,0.5,0.9,0.95), mean=1.67, theta=0.5, alpha=0.5)
#> [1] 0 0 1 4 6
rplindGamma(30, mean=0.5, theta=0.5, alpha=2)
#>  [1] 0 2 2 0 0 0 0 0 0 0 0 3 0 0 1 0 3 0 1 0 0 0 0 0 0 0 0 1 0 0