Poisson-Lindley-Lognormal Distribution

These functions provide density, distribution, quantile, and random generation for the Poisson-Lindley-Lognormal (PLL) Distribution.

Usage

dplindLnorm(
  x,
  mean = 1,
  theta = 1,
  sigma = 1,
  ndraws = 1500,
  log = FALSE,
  hdraws = NULL
)

pplindLnorm(
  q,
  mean = 1,
  theta = 1,
  lambda = NULL,
  sigma = 1,
  ndraws = 1500,
  lower.tail = TRUE,
  log.p = FALSE
)

qplindLnorm(p, mean = 1, theta = 1, sigma = 1, ndraws = 1500, lambda = NULL)

rplindLnorm(n, mean = 1, theta = 1, sigma = 1, ndraws = 1500, lambda = NULL)

Arguments

x

numeric value or vector of values.

mean

mean (>0).

theta

Poisson-Lindley theta parameter (>0).

sigma

lognormal sigma parameter (>0).

ndraws

number of Halton draws.

log

return log-density.

hdraws

optional Halton draws.

q

quantile or vector of quantiles.

lambda

optional lambda parameter (>0).

lower.tail

TRUE returns P[X $$\leq$$ x].

log.p

return log-CDF.

p

probability or vector of probabilities.

n

number of random draws.

Details

The PLL is a 3-parameter count distribution that captures high mass at small y and allows flexible heavy tails.

dplind computes the PLL density. pplind computes the PLL CDF. qplind computes quantiles. rplind generates random draws.

The PMF is: $$ f(y|\mu,\theta,\sigma)=\int_0^\infty \frac{\theta^2\mu^y x^y(\theta+\mu x+y+1)} {(\theta+1)(\theta+\mu x)^{y+2}} \frac{\exp\left(-\frac{\ln^2(x)}{2\sigma^2}\right)} {x\sigma\sqrt{2\pi}}dx $$

Mean: $$ E[y]=\mu=\frac{\lambda(\theta+2)e^{\sigma^2/2}} {\theta(\theta+1)} $$

Halton draws are used to evaluate the integral.

dplindLnorm gives the density, pplindLnorm gives the distribution function, qplindLnorm gives the quantile function, and rplindLnorm generates random deviates.

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

Examples

dplindLnorm(0, mean=0.75, theta=7, sigma=2, ndraws=10)
#> [1] 0.6072274
pplindLnorm(0:10, mean=0.75, theta=7, sigma=2, ndraws=10)
#>  [1] 0.6072274 0.7717043 0.8450268 0.8859278 0.9117652 0.9294614 0.9422986
#>  [8] 0.9520167 0.9596144 0.9657016 0.9706706
qplindLnorm(c(0.1,0.5,0.9), lambda=4.67, theta=7, sigma=2)
#> [1]  0  3 61
rplindLnorm(5, mean=0.75, theta=7, sigma=2)
#> [1] 6 0 0 0 0